On the duality between p-Modulus and probability measures
aa r X i v : . [ m a t h . F A ] A ug On the duality between p -modulusand probability measures L. Ambrosio ∗ , Simone Di Marino † , Giuseppe Savar´e ‡ February 11, 2018
Abstract
Motivated by recent developments on calculus in metric measure spaces ( X, d , m ),we prove a general duality principle between Fuglede’s notion [15] of p -modulus forfamilies of finite Borel measures in ( X, d ) and probability measures with barycenterin L q ( X, m ), with q dual exponent of p ∈ (1 , ∞ ). We apply this general dualityprinciple to study null sets for families of parametric and non-parametric curves in X . In the final part of the paper we provide a new proof, independent of optimaltransportation, of the equivalence of notions of weak upper gradient based on p -podulus ([21], [23]) and suitable probability measures in the space of curves ([6],[7]). Contents
I Duality between modulus and content 5 ( p, m ) -modulus Mod p, m
84 Plans with barycenter in L q ( X, m ) and ( p, m ) -capacity 135 Equivalence between C p, m and Mod p, m ∗ Scuola Normale Superiore, Pisa. email: [email protected] † Scuola Normale Superiore, Pisa. email: [email protected] ‡ Dipartimento di Matematica, Universit`a di Pavia. email: [email protected] I Modulus of families of curves and weak gradients 19 , X ) . . . . . . . . . . . . . . . . . . . . . 216.3 Non-parametric curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 The notion of p -modulus Mod p (Γ) for a family Γ of curves has been introduced by Beurlingand Ahlfors in [2] and then it has been deeply studied by Fuglede in [15], who realizedits significance in Real Analysis and proved that Sobolev W ,p functions f in R n haverepresentatives ˜ f that satisfy˜ f ( γ b ) − ˜ f ( γ a ) = Z ba h∇ f ( γ t ) , γ ′ t i d t for Mod p -almost every absolutely continuous curve γ : [ a, b ] → R n . Recall that if Γ is afamily of absolutely continuous curves, Mod p (Γ) is defined byMod p (Γ) := inf (cid:26)Z R n f p d x : f : R n → [0 , ∞ ] Borel, Z γ f ≥ γ ∈ Γ (cid:27) . (1.1)It is obvious that this definition (as the notion of length) is parametric-free, because thecurves are involved in the definition only through the curvilinear integral R γ f . Furthermore,if γ : I → X , writing the curvilinear integral as R I f ( γ t ) | ˙ γ t | d t , with | ˙ γ | equal to the metricderivative, one realizes immediately that this notion makes sense for absolutely continuouscurves in a general metric space ( X, d ), if we add a reference measure m to minimizethe integral R f p d m . The notion, denoted by Mod p, m ( · ), actually extends to families ofcontinuous curves with finite length, which do have a Lipschitz reparameterization. As in[15], one can even go a step further, realizing that the curvilinear integral in (1.1) can bewritten as Z X f d J γ, where
J γ is a positive finite measure in X , the image under γ of the measure | ˙ γ | L I ,namely J γ ( B ) = Z γ − ( B ) | ˙ γ t | d t ∀ B ∈ B ( X ) (1.2)2here L I stands for the Lebesgue measure on I ). It follows that one can define in asimilar way the notion of p -modulus for families of measures in X .In more recent times, Koskela-Mac Manus [21] and then Shanmugalingham [23] usedthe p -modulus to define the notion of p -weak upper gradient for a function f , namely Borelfunctions g : X → [0 , ∞ ] such that the upper gradient inequality | f ( γ b ) − f ( γ a ) | ≤ Z γ g (1.3)holds along Mod p, m -almost every absolutely continuous curve γ : [ a, b ] → X . This approachleads to a very successful Sobolev space theory in metric measure spaces ( X, d , m ), see forinstance [17, 12] for a very nice account of it.Even more recently, the first and third author and Nicola Gigli introduced (first in [6]for p = 2, and then in [7] for general p ) another notion of weak upper gradient, basedon suitable classes of probability measures on curves, described more in detail in the finalsection of this paper. Since the axiomatization in [6] is quite different and sensitive to pa-rameterization, it is a surprising fact that the two approaches lead essentially to the sameSobolev space theory (see Remark 5.12 of [6] for a more detailed discussion, also in connec-tion with Cheeger’s approach [13], and Section 9 of this paper). We say essentially because,strictly speaking, the axiomatization of [6] is invariant (unlike Fuglede’s approach) undermodification of f in m -negligible sets and thus provides only Sobolev regularity and notabsolute continuity along almost every curve; however, choosing properly representativesin the Lebesgue equivalence class, the two Sobolev spaces can be identified.Actually, as illustrated in [6], [8], [16] (see also the more recent work [10], in connec-tion with Rademacher’s theorem and Cheeger’s Lipschitz charts), differential calculus andsuitable notions of tangent bundle in metric measure spaces can be developed in a quitenatural way using probability measures in the space of absolutely continuous curves.With the goal of understanding deeper connections between the Mod p, m and the prob-abilistic approaches, we show in this paper that the theory of p -modulus has a “dual”point of view, based on suitable probability measures π in the space of curves; the maindifference with respect to [6] is that, as it should be, the curves here are non-parametric,namely π should be rather thought as measures in a quotient space of curves. Actually,this and other technical aspects (also relative to tightness, since much better compactnessproperties are available at the level of measures) are simplified if we consider p -modulusof families of measures in M + ( X ) (the space of all nonnegative and finite Borel measureson X ), rather than p -modulus of families of curves: if we have a family Γ of curves, wecan consider the family Σ = J (Γ) and derive a representation formula for Mod p, m (Γ), seeSection 7. Correspondingly, π will be a measure on the Borel subsets of M + ( X ).For this reason, in Part I of this paper we investigate the duality at this level of general-ity, considering a family Σ of measures in M + ( X ). Assuming only that ( X, d ) is completeand separable and m is finite, we prove in Theorem 5.1 that for all Borel sets Σ ⊂ M + ( X )(and actually in the more general class of Souslin sets) the following duality formula holds: (cid:2) Mod p, m (Σ) (cid:3) /p = sup η η (Σ) c q ( η ) = sup η (Σ)=1 c q ( η ) , p + 1 q = 1 . (1.4)3ere the supremum in the right hand side runs in the class of Borel probability measures η in M + ( X ) with barycenter in L q ( X, m ), so thatthere exists g ∈ L q ( X, m ) s.t. Z µ ( A ) d η ( µ ) = Z A g d m ∀ A ∈ B ( X );the constant c q ( η ) is then defined as the L q ( X, m ) norm of the “barycenter” g . A byproductof our proof is the fact that Mod p, m is a Choquet capacity in M + ( X ), see Theorem 5.1.In addition, we can prove in Corollary 5.2 existence of maximizers in (1.4) and obtain outof this necessary and sufficient optimality conditions, both for η and for the minimal f involved in the definition of p -modulus analogous to (1.1). See also Remark 3.3 for a simpleapplication of these optimality conditions involving pairs ( µ, f ) on which the constraint issaturated, namely R X f d µ = 1.We are not aware of other representation formulas for Mod p, m , except in special cases:for instance in the case of the family Γ of curves connecting two disjoint compact sets K , K of R n , the modulus in (1.1) equals (see [24] and also [20] for the extension to metricmeasure spaces, as well as [1] for related results) the capacity C p ( K , K ) := inf (cid:26)Z R n |∇ u | p d x : u ≡ K , u ≡ K (cid:27) . In the conformal case p = n , it can be also proved that C n ( K , K ) − / ( n − equals Mod n/ ( n − (Σ),where Σ is the family of the Hausdorff measures H n − S , with S separating K from K (see [25]).In the second part of the paper, after introducing in Section 6 the relevant space ofcurves AC q ([0 , X ) and a suitable quotient space C ( X ) of non-parametric nonconstantcurves, we show how the basic duality result of Part I can be read in terms of measures andmoduli in spaces of curves. For non-parametric curves this is accomplished in Section 7,mapping curves in X to measures in X with the canonical map J in (1.2); in this case, thecondition of having a barycenter in L q ( X, m ) becomes (cid:12)(cid:12)(cid:12)(cid:12)Z Z f ( γ t ) | ˙ γ t | d t d π ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k f k L p ( X, m ) ∀ f ∈ C b ( X ) . (1.5)Section 8 is devoted instead to the case of parametric curves, where the relevant mapcurves-to-measures is M γ ( B ) := L ( γ − ( B )) ∀ B ∈ B ( X ) . In this case the condition of having a parametric barycenter in L q ( X, m ) becomes (cid:12)(cid:12)(cid:12)(cid:12)Z Z f ( γ t ) d t d π ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k f k L p ( X, m ) ∀ f ∈ C b ( X ) . (1.6)The parametric barycenter can of course be affected by reparameterizations; a key result,stated in Theorem 8.5, shows that suitable reparameterizations improve the parametric4arycenter from L q ( X, m ) to L ∞ ( X, m ). Then, in Section 9 we discuss the notion of null setof curves according to [6] and [7] (where (1.6) is strengthened by requiring (cid:12)(cid:12)R f ( γ t ) d π ( γ ) (cid:12)(cid:12) ≤ C k f k L ( X, m ) for all t , for some C independent of t ) and, under suitable invariance andstability assumptions on the set of curves, we compare this notion with the one based on p -modulus. Eventually, in Section 10 we use there results to prove that if a Borel function f : X → R has a continuous representative along a collection Γ of the set AC ∞ ([0 , X ) ofthe Lipschitz parametric curves with Mod p, m (cid:0) M (AC ∞ ([0 , X ) \ Γ) (cid:1) = 0, then it is possibleto find a distinguished m -measurable representative ˜ f such that m ( { f = ˜ f } ) = 0 and ˜ f is absolutely continuous along Mod p, m -a.e.-nonparametric curve. By using these resultsto provide a more direct proof of the equivalence of the two above mentioned notions ofweak upper gradient, where different notions of null sets of curves are used to quantifyexceptions to (1.3).For the reader’s convenience we collect in the next table and figure the main notationused, mostly in the second part of the paper.Main notation L p + ( X, m ) Borel nonnegative functions f : X → [0 , ∞ ] with R X f p d m < ∞ L p ( X, m ) Lebesgue space of p -summable m -measurable functions ℓ ( γ ) Length of a parametric curve γ AC q ([0 , X ) Space of parametric curves γ : [0 , → X with q -integrable metric speedAC ([0 , X ) Space of parametric curves with positive speed L -a.e. in (0 , ∞ c ([0 , X ) Space of parametric curves with positive and constant speed k Embedding of (cid:8) γ ∈ AC([0 , X ) : ℓ ( γ ) > (cid:9) into AC ∞ c ([0 , X ) C ( X ) Space of non-parametric and nonconstant curves, see Definition 6.5 i Embedding of (cid:8) γ ∈ AC([0 , X ) : ℓ ( γ ) > (cid:9) in C ( X ) j Embedding of C ( X ) into AC ∞ c ([0 , X )AC ∞ c ([0 , X ) C ( X ) { γ ∈ AC([0 , X ) : ℓ ( γ ) > } C([0 , X ) M + ( X ) π C j ˜ J k i JM Acknowledgement.
The first author acknowledges the support of the ERC ADG GeMeThNES.The first and third author have been partially supported by PRIN10-11 grant from MIURfor the project Calculus of Variations. All authors thank the reviewer, whose detailedcomments led to an improvement of the manuscript.5 art I
Duality between modulus and content
In a topological Hausdorff space (
E, τ ), we denote by P ( E ) the collection of all subsets of E , by F ( E ) (resp. K ( E )) the collection of all closed (resp. compact) sets of E , by B ( E )the σ -algebra of Borel sets of E . We denote by C b ( E ) the space of bounded continuousfunctions on ( E, τ ), by M + ( E ), the set of σ -additive measures µ : B ( E ) → [0 , ∞ ), by P ( E ) the subclass of probability measures. For a set F ⊂ E and µ ∈ M + ( E ) we shallrespectively denote by χ F : E → { , } the characteristic function of F and by µ F the measure χ F µ , if F is µ -measurable. For a Borel map L : E → F we shall denoteby L ♯ : M + ( E ) → M + ( F ) the induced push-forward operator between Borel measures,namely L ♯ µ ( B ) := µ (cid:0) L − ( B ) (cid:1) ∀ µ ∈ M + ( E ) , B ∈ B ( F ) . We shall denote by N = { , , . . . } the natural numbers, by L the Lebesgue measure onthe real line. Recall that (
E, τ ) is said to be Polish if there exists a distance ρ in E which induces thetopology τ such that ( E, ρ ) is complete and separable. Notice that the inclusion of M + ( E )in (C b ( E )) ∗ may be strict, because we are not making compactness or local compactnessassumptions on ( E, τ ). Nevertheless, if (
E, τ ) is Polish we can always endow M + ( E ) witha Polish topology w -C b ( E ) whose convergent sequences are precisely the weakly conver-gent ones, i.e. sequences convergent in the duality with C b ( E ). Obviously this Polishtopology is unique. A possible choice, which can be easily adapted from the correspond-ing Kantorovich-Rubinstein distance on P ( E ) (see e.g. [11, § ρ KR ( µ, ν ) := sup n(cid:12)(cid:12)(cid:12) Z E f d µ − Z E f d ν (cid:12)(cid:12)(cid:12) : f ∈ Lip b ( E ) , sup E | f | ≤ , | f ( x ) − f ( y ) | ≤ ρ ( x, y ) ∀ x, y ∈ E o . Denote by N ∞ the collection of all infinite sequences of natural numbers and by N ∞ thecollection of all finite sequences ( n , . . . , n i ), with i ≥ n i natural numbers. Let A ⊂ P ( E ) containing the empty set (typical examples are, in topological spaces ( E, τ ),the classes F ( E ), K ( E ), B ( E )). We call table of sets in A a map C associating to eachfinite sequence ( n , . . . , n i ) ∈ N ∞ a set C ( n ,...,n i ) ∈ A .6 efinition 2.1 ( A -analytic sets) A set S ⊂ E is said to be A -analytic if there existsa table C of sets in A such that S = [ ( n ) ∈ N ∞ ∞ \ i =0 C ( n ,...,n i ) . Recall that, in a topological space (
E, τ ), B ( E )-analytic sets are universally measurable [11, Theorem 1.10.5]: this means that they are σ -measurable for any σ ∈ M + ( E ). Definition 2.2 (Souslin and Lusin sets)
Let ( E, τ ) be an Hausdorff topological space. S ∈ P ( E ) is said to be a Souslin (resp. Lusin) set if it is the image of a Polish spaceunder a continuous (resp. continuous and injective) map. Even though the Souslin and Lusin properties for subsets of a topological space areintrinsic, i.e. they depend only on the induced topology, we will often use the diction“ S Suslin subset of E ” and similar to emphasize the ambient space; the Borel property,instead, is not intrinsic, since S ∈ B ( S ) if we endow S with the induced topology. Besidesthe obvious stability with respect to transformations through continuous (resp. continuousand injective) maps, the class of Souslin (resp. Lusin) sets enjoys nice properties, detailedbelow. Proposition 2.3
The following properties hold:(i) In a Hausdorff topological space ( E, τ ) , Souslin sets are F ( E ) -analytic;(ii) if ( E, τ ) is a Souslin space (in particular if it is a Polish or a Lusin space), thenotions of Souslin and F ( E ) -analytic sets concide and in this case Lusin sets areBorel and Borel sets are Souslin;(iii) if E , F are Souslin spaces and f : E → F is a Borel injective map, then f − is Borel;(iv) if E , F are Souslin spaces and f : E → F is a Borel map, then f maps Souslin setsto Souslin sets.Proof. We quote [11] for all these statements: (i) is proved in Theorem 6.6.8; in con-nection with (ii), the equivalence between Souslin and F ( E )-analytic sets is proved inTheorem 6.7.2, the fact that Borel sets are Souslin in Corollary 6.6.7 and the fact thatLusin sets are Borel in Theorem 6.8.6; finally, (iii) and (iv) are proved in Theorem 6.7.3. (cid:3) Since in Polish spaces (
E, τ ) we have at the same time tightness of finite Borel measuresand coincidence of Souslin and F ( E )-analytic sets, the measurability of B ( E )-analytic setsyields in particular that σ ( B ) = sup { σ ( K ) : K ∈ K ( E ) , K ⊂ B } for all B ⊂ E Souslin, σ ∈ M + ( E ). (2.1)We will need a property analogous to (2.1) for capacities [14], whose definition is recalledbelow. 7 efinition 2.4 (Capacity) A set function I : P ( E ) → [0 , ∞ ] is said to be a capacity if: • I is nondecreasing and, whenever ( A n ) ⊂ P ( E ) is nondecreasing, the following holds lim n →∞ I ( A n ) = I ∞ [ n =0 A n ! ; • if ( K n ) ⊂ K ( E ) is nonincreasing, the following holds: lim n →∞ I ( K n ) = I ∞ \ n =0 K n ! . A set B ⊂ E is said to be I -capacitable if I ( B ) = sup K ∈ K ( E ) , K ⊂ B I ( K ) . Theorem 2.5 (Choquet) ([14, Thm 28.III])
Every K ( E ) -analytic set is capacitable. ( p, m ) -modulus Mod p, m In this section (
X, τ ) is a topological space and m is a fixed Borel and nonnegative referencemeasure, not necessarily finite or σ -finite.Given a power p ∈ [1 , ∞ ), we set L p + ( X, m ) := (cid:26) f : X → [0 , ∞ ] : f Borel, Z X f p d m < ∞ (cid:27) . (3.1)We stress that, unlike L p ( X, m ), this space is not quotiented under any equivalence relation;however we will keep using the notation k f k p := (cid:18)Z X | f | p d m (cid:19) /p as a seminorm on L p + ( X, m ) and a norm in L p ( X, m ).Given Σ ⊂ M + we define (with the usual convention inf ∅ = ∞ )Mod p, m (Σ) := inf (cid:26)Z X f p d m : f ∈ L p + ( X, m ) , Z X f d µ ≥ µ ∈ Σ (cid:27) , (3.2)Mod p, m ,c (Σ) := inf (cid:26)Z X f p d m : f ∈ C b ( X, [0 , ∞ )) , Z X f d µ ≥ µ ∈ Σ (cid:27) . (3.3)Equivalently, if 0 < Mod p, m (Σ) ≤ ∞ , we can say that Mod p, m (Σ) − is the least number ξ ∈ [0 , ∞ ) such that the following is true (cid:18) inf µ ∈ Σ Z X f d µ (cid:19) p ≤ ξ Z X f p d m for all f ∈ L p + ( X, m ) , (3.4)8nd similarly there is also an equivalent definition for Mod p, m ,c (Σ) − .Notice that the infimum in (3.3) is unchanged if we restrict the minimization to nonnegativefunctions f ∈ C b ( X ). As a consequence, since the finiteness of m provides the inclusion ofthis class of functions in L p + ( X, m ), we get Mod p, m ,c (Σ) ≥ Mod p, m (Σ) whenever m is finite.Also, whenever Σ contains the null measure, we have Mod p, m ,c (Σ) ≥ Mod p, m (Σ) = ∞ . Definition 3.1 (
Mod p, m -negligible sets) A set Σ ⊂ M + ( X ) is said to be Mod p, m -negligibleif Mod p, m (Σ) = 0 . A property P on M + ( X ) is said to hold Mod p, m -a.e. if the set { µ ∈ M + ( X ) : P ( µ ) fails } is Mod p, m -negligible. With this terminology, we can also writeMod p, m (Σ) = inf (cid:26)Z X f p d m : Z X f d µ ≥ p, m -a.e. µ ∈ Σ (cid:27) . (3.5)We list now some classical properties that will be useful in the sequel, most them are wellknown and simple to prove, but we provide complete proofs for the reader’s convenience. Proposition 3.2
The set functions A ⊂ M + ( X ) Mod p, m ( A ) , A ⊂ M + ( X ) Mod p, m ,c ( A ) satisfy the following properties:(i) both are monotone and their /p -th power is subadditive;(ii) if g ∈ L p + ( X, m ) then R X g d µ < ∞ for Mod p, m -almost every µ ; conversely, if Mod p, m ( A ) =0 then there exists g ∈ L p + ( X, m ) such that R X g d µ = ∞ for every µ ∈ A .(iii) if ( f n ) ⊂ L p + ( X, m ) converges in L p ( X, m ) seminorm to f ∈ L p + ( X, m ) , there exists asubsequence ( f n ( k ) ) such that Z X f n ( k ) d µ → Z X f d µ Mod p, m -a.e. in M + ( X ); (3.6) (iv) if p > , for every Σ ⊂ M + ( X ) with Mod p, m (Σ) < ∞ there exists f ∈ L p + ( X, m ) ,unique up to m -negligible sets, such that R X f d µ ≥ p, m -a.e. on Σ and k f k pp =Mod p, m (Σ) ;(v) if p > and A n are nondecreasing subsets of M + ( X ) then Mod p, m ( A n ) ↑ Mod p, m ( ∪ n A n ) ;(vi) if K n are nonincreasing compact subsets of M + ( X ) then Mod p, m ,c ( K n ) ↓ Mod p, m ,c ( ∩ n K n ) .(vii) Let A ⊂ M + ( X ) , F : A → (0 , ∞ ) be a Borel map, and B = (cid:8) F ( µ ) µ : µ ∈ A (cid:9) . If Mod p, m ( A ) = 0 then Mod p, m ( B ) = 0 as well. roof. (i) Monotonicity is trivial. For the subadditivity, if we take R X f d µ ≥ A and R X g d µ ≥ B , then R X ( f + g ) d µ ≥ A ∪ B , hence Mod p, m ( A ∪ B ) /p ≤ k f + g k p ≤k f k p + k g k p . Minimizing over f and g we get the subadditivity.(ii) Let us consider the set where the property fails:Σ g = (cid:26) µ ∈ M + ( X ) : Z X g d µ = ∞ (cid:27) . Then it is clear that Mod p, m (Σ g ) ≤ k g k pp but Σ g = Σ λg for every λ > g is Mod p, m -negligible. Conversely, if Mod p, m ( A ) = 0 for every n ∈ N we can find g n ∈ L p + ( X, m ) with R X g n d µ ≥ µ ∈ A and R X g pn ≤ − np . Thus g := P n g n satisfies the required properties.(iii) Let f n ( k ) be a subsequence such that k f − f n ( k ) k p ≤ − k so that if we set g ( x ) = ∞ X k =1 | f ( x ) − f n ( k ) ( x ) | we have that g ∈ L p + ( X, m ) and k g k p ≤
1; in particular we have, for (ii) above, that R X g d µ is finite for Mod p, m -almost every µ . For those µ we get ∞ X k =1 Z X | f − f n ( k ) | d µ < ∞ and thus we get (3.6).(iv) Since we can use (3.5) to compute Mod p, m (Σ), we obtain from (ii) and (iii) thatthe class of admissible functions f is a convex and closed subset of the Lebesgue space L p .Hence, uniqueness follows by the strict convexity of the L p norm.(v) By the monotonicity, it is clear that Mod p, m ( A n ) is an increasing sequence and thatMod p, m ( ∪ n A n ) ≥ lim Mod p, m ( A n ) =: C . If C = ∞ there is nothing to prove, otherwise,we need to show that Mod p, m ( ∪ n A n ) ≤ C ; let ( f n ) ⊂ L p + ( X, m ) be a sequence of functionssuch that R X f n d µ ≥ A n and k f n k pp ≤ Mod p, m ( A n ) + n . In particular we get thatlim sup n k f n k pp = C < ∞ and so, possibly extracting a subsequence, we can assume that f n weakly converge to some f ∈ L p + ( X, m ). By Mazur lemma we can find convex combinationsˆ f n = ∞ X k = n λ k,n f k such that ˆ f n converge strongly to f in L p ( X, m ); furthermore we have that R X f k d µ ≥ A n if k ≥ n and so Z X ˆ f n d µ = ∞ X k = n λ k,n Z X f k d µ ≥ A n .
10y (iii) in this proposition we obtain a subsequence n ( k ) and a Mod p, m -negligible setΣ ⊂ M + ( X ) such that R X ˆ f n ( k ) d µ → R X f d µ outside Σ; in particular R X f d µ ≥ ∪ n A n \ Σ. Then, by the very definition of Mod p, m -negligible set, for every ε > g ε ∈ L p + ( X, m ) such that k g ε k pp ≤ ε and R X g ε d µ ≥ R X ( f + g ε ) d µ ≥ ∪ n A n andMod p, m ( ∪ n A n ) /p ≤ k g ε + f k p ≤ k g ε k p + k f k p ≤ ε /p + lim inf k f n k p ≤ ε /p + C /p . Letting ε → p -th power the inequality Mod p, m ( ∪ n A n ) ≤ sup n Mod p, m ( A n )follows.(vi) Let K = ∩ n K n . As before, by the monotonicity we get Mod p, m ,c ( K ) ≤ Mod p, m ,c ( K n )and so calling C the limit of Mod p, m ,c ( K n ) as n goes to infinity, we only have to proveMod p, m ,c ( K ) ≥ C . First, we deal with the case Mod p, m ,c ( K ) >
0: using the equivalentdefinition, let φ ε ∈ C b ( X ) be such that k φ ε k p = 1 andinf µ ∈ K Z X φ ε d µ ≥ p, m ,c ( K ) /p − ε. By the compactness of K and of K n , it is clear that the infimum above is a minimum andthat min K n R X φ ε dµ → min K R X φ ε d µ , so that1 C /p = lim n →∞ p, m ,c ( K n ) /p ≥ lim n →∞ min µ ∈ K n Z X φ ε d µ ≥ p, m ,c ( K ) /p − ε. The case Mod p, m ,c ( K ) = 0 is the same, taking φ M ∈ C b ( X ) such that k φ M k p = 1 and R X φ M dµ ≥ M on K and then letting M → ∞ .(vii) Since Mod p, m ( A ) = 0, by (ii) we find g ∈ L p + ( X, m ) such that R X g d µ = ∞ forevery µ ∈ A : this yields R X g d (cid:0) F ( µ ) µ (cid:1) = ∞ for every µ ∈ A , showing that Mod p, m ( B ) = 0. (cid:3) Remark 3.3
In connection with Proposition 3.2(iv), in general the constraint R X f dµ ≥ f , namely the strict inequality can occur for a subset Σ with positive ( p, m )-modulus. For instance, if X = [0 ,
1] and m is the Lebesgue measure,then Mod p, m (cid:0) { L [0 ,
12 ] , L [ 12 , , L [0 , } (cid:1) = 2 p and f ≡ , but R X f d m = 2. However, we will prove using the duality formula Mod p, m = C pp, m thatone can always find a subset Σ ′ ⊂ Σ (in the example above Σ \ Σ ′ = { L [0 , } ) withthe same ( p, m )-modulus satisfying R X f dµ = 1 for all µ ∈ Σ ′ , see the comment made afterCorollary 5.2.On the other hand, if the measures in Σ are non-atomic, using just the definition of p -modulus, one can find instead a family Σ ′ of smaller measures with the same modulus asΣ on which the constraint is saturated: suffices to find, for any µ ∈ Σ, a smaller measure µ ′ (a subcurve, in the case of measures associated to curves) satisfying R X f dµ ′ = 1. In theprevious example the two constructions lead to the same result, but the two proceduresare conceptually quite different. 11nother important property is the tightness of Mod p, m in M + ( X ): it will play a crucialrole in the proof of Theorem 5.1 to prove the inner regularity of Mod p, m for arbitrarySouslin sets. Lemma 3.4 (Tightness of
Mod p, m ) If ( X, τ ) is Polish and m ∈ M + ( X ) , for every ε > there exists E ε ⊂ M + ( X ) compact such that Mod p, m ( E cε ) ≤ ε .Proof. Since (
X, τ ) is Polish, by Ulam theorem we can find a nondecreasing family of sets K n ∈ K ( X ) such that m ( K cn ) → . We claim the existence of δ n ↓ E k = { µ ∈ M + ( X ) : µ ( X ) ≤ k and µ ( K cn ) ≤ δ n ∀ n ≥ k } , then E k is compact and Mod p, m ( E ck ) → k goes to infinity. First of all it is easy to seethat the family { E k } is compact by Prokhorov theorem, because it is clearly tight.To evaluate Mod p, m ( E ck ) we have to build some functions. Let m n = m ( K cn ), assumewith no loss of generality that m n > n , set a n = ( √ m n + √ m n +1 ) − /p and note thatthis latter sequence is nondecreasing and diverging to + ∞ ; let us now define the functions f k ( x ) := x ∈ K k ,a n if x ∈ K n +1 \ K n and n ≥ k, + ∞ otherwise.Now we claim that if we put δ n = a − n in the definition of the E k ’s we will have Mod p, m ( E ck ) →
0: in fact, if µ ∈ E ck then we have either µ ( X ) > k or µ ( K cn ) > δ n for some n ≥ k . In eithercase the integral of the function f k + k with respect to µ is greater or equal to 1: • if µ ( X ) > k then Z X (cid:18) f k + 1 k (cid:19) d µ ≥ Z X k d µ ≥ • if µ ( K cn ) > δ n for some n ≥ k we have that Z X (cid:18) f k + 1 k (cid:19) d µ ≥ Z K cn f k d µ ≥ Z K cn a n d µ > δ n a n = 1 . So we have that Mod p, m ( E ck ) ≤ k f k + k k pp ≤ ( k f k k p + k /k k p ) p . But Z X f pk d m = ∞ X n = k Z K n +1 \ K n a pn d m = ∞ X n = k m n − m n +1 √ m n + √ m n +1 = ∞ X n = k ( √ m n − √ m n +1 ) = √ m k , and so we have Mod p, m ( E ck ) ≤ (cid:16) ( m k ) / (2 p ) + ( m ( X )) /p /k (cid:17) p → (cid:3) Plans with barycenter in L q ( X, m ) and ( p, m ) -capacity In this section (
X, τ ) is Polish and m ∈ M + ( X ) is a fixed reference measure. We will endow M + ( X ) with the Polish structure making the maps µ R X f dµ , f ∈ C b ( X ), continuous,as described in Section 2. Definition 4.1 (Plans with barycenter in L q ( X, m ) ) Let q ∈ (1 , ∞ ] , p = q ′ . We saythat a Borel probability measure η on M + ( X ) is a plan with barycenter in L q ( X, m ) if thereexists c ∈ [0 , ∞ ) such that Z Z X f d µ d η ( µ ) ≤ c k f k p ∀ f ∈ L p + ( X, m ) . (4.1) If η is a plan with barycenter in L q ( X, m ) , we call c q ( η ) the minimal c in (4.1) . Notice that c q ( η ) = 0 iff η is the Dirac mass at the null measure in M + ( X ). We alsoused implicitly in (4.1) (and in the sequel it will be used without further mention) thefact that µ R X f d µ is Borel whenever f ∈ L p + ( X, m ). The proof can be achieved by astandard monotone class argument.An equivalent definition of the class plans with barycenter in L q ( X, m ), which explainsalso the terminology we adopted, is based on the requirement that the barycenter Borelmeasure µ := Z µ d η ( µ ) (4.2)is absolutely continuous w.r.t. m and with a density ρ in L q ( X, m ). Moreover, c q ( η ) = k ρ k q . (4.3)Indeed, choosing f = χ A in (4.1) gives µ ( A ) ≤ ( m ( A )) /p , hence the Radon-Nikodymtheorem provides the representation µ = ρ m for some ρ ∈ L ( X, m ). Then, (4.1) oncemore gives Z X ρf d m ≤ c k f k p ∀ f ∈ L p ( X, m )and the duality of Lebesgue spaces gives ρ ∈ L q ( X, m ) and k ρ k q ≤ c . Conversely, if µ hasa density in L q ( X, m ), we obtain by H¨older’s inequality that (4.1) holds with c = k ρ k q .Obviously, (4.1) still holds with c = c q ( η ) for all f ∈ C b ( X ), not necessarily nonnegative,when η is a plan with good barycenter in L q ( X, m ). Actually the next proposition showsthat we need only to check the inequality (4.1) for f ∈ C b ( X ) nonnegative. Proposition 4.2
Let η be a probability measure on M + ( X ) such that Z Z X f d µ d η ( µ ) ≤ c k f k p for all f ∈ C b ( X ) nonnegative (4.4) for some c ≥ . Then (4.4) holds, with the same constant c , also for every f ∈ L p + ( X, m ) . roof. It suffices to remark that (4.4) gives Z X f d µ ≤ c k f k p ∀ f ∈ C b ( X ) , with µ defined in (4.2). Again the duality of Lebesgue spaces provides ρ ∈ L q ( X, m ) with k ρ k q ≤ c satisfying R X f ρ d m = R X f d µ for all f ∈ C b ( X ), hence µ = ρ m . (cid:3) There is a simple duality inequality, involving the minimization in (3.2) and a maxi-mization among all η ’s with barycenter in L q ( X, m ). To see it, let’s take f ∈ L p + ( X, m )such that R f d µ ≥ ⊂ M + ( X ). Then, if Σ is universally measurable we may takeany plan η with barycenter in L q ( X, m ) to obtain η (Σ) ≤ Z Z X f d µ d η ( µ ) ≤ c q ( η ) k f k p . (4.5)In particular we haveMod p, m (Σ) = 0 = ⇒ η (Σ) = 0 for all η with barycenter in L q ( X, m ) . (4.6)In addition, taking in (4.5) the infimum over all the f ∈ L p + ( X, m ) such that R f d µ ≥ η with barycenter in L q ( X, m ) and c q ( η ) >
0, we findsup c ( η ) > η (Σ) c q ( η ) ≤ Mod p, m (Σ) /p . (4.7)The inequality (4.7) motivates the next definition. Definition 4.3 ( ( p, m ) -content) If Σ ⊂ M + ( X ) is a universally measurable set we define C p, m (Σ) := sup c q ( η ) > η (Σ) c q ( η ) . (4.8) By convention, we set C p, m (Σ) = ∞ if ∈ Σ . A first important implication of (4.7) is that for any family F of plans η with barycenterin L q ( X, m ) C := sup { c q ( η ) : η ∈ F } < ∞ = ⇒ F is tight. (4.9)Indeed, η ( E cε p ) ≤ εc q ( η ) ≤ Cε , where the E ε ⊂ M + ( X ) are the compact sets provided byLemma 3.4. This allows to prove existence of optimal η ’s in (4.8). Lemma 4.4
Let Σ ⊂ M + ( X ) be a universally measurable set such that C p, m (Σ) > and sup Σ µ ( X ) < ∞ . Then there exists an optimal plan η with barycenter in L q ( X, m ) in (4.8) ,and any optimal plan is concentrated on Σ . In particular C p, m (Σ) = η (Σ) c q ( η ) = 1 c q ( η ) . roof. First we claim that the supremum in (4.7) can be restricted to the plans withbarycenter in L q ( X, m ) concentrated on Σ. Indeed, given any admissible η with η (Σ) > η ′ = ( η (Σ)) − χ Σ η we obtain another plan with barycenter in L q ( X, m ) satisfying η ′ (Σ) = 1 and Z Z X f d µ d η ′ ( µ ) = 1 η (Σ) Z Σ Z X f d µ d η ( µ ) ≤ η (Σ) Z Z X f d µ d η ( µ ) ≤ c q ( η ) η (Σ) k f k p for all f ∈ L p + ( X, m ). In particular the definition of c q ( η ′ ) gives c q ( η ′ ) ≤ c q ( η ) η (Σ) , and proves our claim. The same argument proves that η ′ = η whenever η is a maximizer.Now we know that C p, m (Σ) = sup η (Σ)=1 c q ( η ) , where the supremum is made over plans with barycenter in L q ( X, m ). We take a maximiz-ing sequence ( η k ); for this sequence we have that c q ( η k ) ≤ C , so that ( η k ) is tight by (4.9).Assume with no loss of generality that η k weakly converges to some η , that is clearly aprobability measure in M + ( X ). To see that η is a plan with barycenter in L q ( X, m ) andthat c q ( η ) is optimal, we notice that the continuity and boundedness of µ R X f d µ inbounded sets of M + ( X ) for f ∈ C b ( X ) gives Z Z X f d µ d η ( µ ) = lim k →∞ Z Z X f d µ d η k ( µ ) ≤ lim k →∞ c q ( η k ) k f k p , so that Z Z X f d µ d η ( µ ) ≤ C p, m (Σ) k f k p ∀ f ∈ C b ( X ) . The thesis follows from Proposition 4.2. (cid:3) C p, m and Mod p, m In the previous two sections, under the standing assumptions (
X, τ ) Hausdorff topologicalspace (Polish in the case of C p, m ), µ ∈ M + ( X ) and p ∈ [1 , ∞ ), we introduced a p -modulusMod p, m and a p -content C p, m , proving the direct inequalities (see (4.7)) C pp, m ≤ Mod p, m ≤ Mod p, m ,c on Souslin subsets of M + ( X ).Under the same assumptions on ( X, τ ) and m ∈ M + ( X ), our goal in this section is thefollowing result: 15 heorem 5.1 Let ( X, τ ) be a Polish topological space and p > . Then Mod p, m is aChoquet capacity in M + ( X ) , every Souslin set Σ ⊂ M + ( X ) is capacitable and satisfies Mod p, m (Σ) /p = C p, m (Σ) . If moreover Σ is also compact we have Mod p, m (Σ) = Mod p, m ,c (Σ) .Proof. We split the proof in two steps: • first, prove that Mod p, m ,c (Σ) /p ≤ C p, m (Σ) if Σ is compact, so that in particularMod /pp, m = C p, m on compact sets; • then, prove that Mod p, m and C p, m are inner regular, and deduce that Mod /pp, m = C p, m on Souslin sets.The two steps together yield Mod p, m = Mod p, m ,c on compact sets, hence we can use Propo-sition 3.2(v,vi) to obtain that Mod p, m is a Choquet capacity in M + ( X ). Step 1.
Assume that Σ ⊂ M + ( X ) is compact. In particular sup Σ µ ( X ) is finite and so wehave that the linear map Φ : C b ( X ) → C(Σ) = C b (Σ) given by f Φ f ( µ ) := Z X f d µ is a bounded linear operator.If Σ contains the null measure there is nothing to prove, because Mod p, m ,c (Σ) = ∞ by definition and C p, m (Σ) = ∞ by convention. If not, by compactness, we obtain that ε := inf Σ µ ( X ) >
0, so that taking f ≡ ε − in (3.3) we obtain Mod p, m ,c (Σ) < ∞ . We canalso assume that Mod p, m ,c (Σ) >
0, otherwise there is nothing to prove.Our first step is the construction of a plan η with barycenter in L q ( X, m ) concentratedon Σ. By the equivalent definition analogous to (3.4) for Mod p, m ,c , the constant ξ =Mod p, m ,c (Σ) − /p satisfies inf µ ∈ Σ Φ f ( µ ) ≤ ξ k f k p ∀ f ∈ C( X ) . (5.1)Denoting by v = v ( µ ) a generic element of C(Σ), we will now consider two functions onC(Σ): F ( v ) = inf {k f k p : f ∈ C b ( X ) , Φ f ≥ v on Σ } F ( v ) = min { v ( µ ) : µ ∈ Σ } . The following properties are immediate to check, using the linearity of f Φ f for the firstone and (5.1) for the third one: • F is convex; • F is continuous and concave; • F ≤ ξ · F . 16ith these properties, standard Banach theory gives us a continuous linear functional L ∈ (C(Σ)) ∗ such that F ( v ) ≤ L ( v ) ≤ ξ · F ( v ) ∀ v ∈ C(Σ) . (5.2)For the reader’s convenience we detail the argument: first we apply the geometric form ofthe Hahn-Banach theorem in the space C(Σ) × R to the convex sets A = { F ( v ) > t } and B = { F ( v ) ≤ t/ξ } , where the former is also open, to obtain a continuous linear functional G in C(Σ) × R such that G ( v, t ) < G ( w, s ) whenever F ( v ) > t , F ( w ) ≤ s/ξ .Representing G ( v, t ) as H ( v ) + βt for some H ∈ (C(Σ)) ∗ and β ∈ R , the inequality reads H ( v ) + βt < H ( w ) + βs whenever F ( v ) > t , F ( w ) ≤ s/ξ .Since F and F are real-valued, β >
0; we immediately get F ≤ ( γ − H ) /β ≤ ξF , with γ := sup H ( v ) + βF ( v ). On the other hand, F (0) = F (0) = 0 implies γ = 0, so that wecan take L = − H/β in (5.2).In particular from (5.2) we get that if v ≥ L ( v ) ≥ F ( v ) ≥ η in Σ representing L : L ( v ) = Z Σ v ( µ ) d η ∀ v ∈ C(Σ) . Furthermore this measure can’t be null since (here is the function identically equal to 1). η (Σ) = L ( ) ≥ F ( ) = 1 , and so η (Σ) ≥
1. Now we claim that η is a plan with barycenter in L q ( X, m ); first we provethat η (Σ) ≤
1, so that η will be a probability measure. In fact, we know F ( v ) η (Σ) ≤ L ( v )because v ≥ F ( v ) on Σ, and then F ( v ) η (Σ) ≤ ξF ( v ) . In particular, inserting in this inequality v = Φ φ with φ ∈ C b ( X ), we obtaininf Σ Φ φ ≤ ξ η (Σ) k φ k p and so Mod p, m ,c (Σ) ≥ ( η (Σ) /ξ ) p = η (Σ) p Mod p, m ,c (Σ), which implies η (Σ) ≤
1. Now wehave that Z Σ (cid:18)Z X f d µ (cid:19) d η = L (Φ f ) ≤ ξ · F (Φ f ) ≤ ξ · k f k p ∀ f ∈ C b ( X ) (5.3)17nd so, by Proposition 4.2, this inequality is true for every f ∈ L p + ( X, m ), showing that η is a plan with barycenter in L q ( X, m ); as a byproduct we gain also that c q ( η ) ≤ ξ thatgives us, that C p, m (Σ) ≥ Mod p, m ,c (Σ) /p , thus obtaining that C p, m (Σ) = Mod p, m (Σ) /p = Mod p, m ,c (Σ) /p . Step 2.
Now we will prove that Mod p, m and C p, m are both inner regular, namely theirvalue on Souslin sets is the supremum of their value on compact subsets. Inner regularityand equality on compact sets yield C p, m ( B ) = Mod p, m ( B ) /p on every Souslin subset B of M + ( X ).Mod p, m is inner regular. Proposition 3.2(v,vi) and the fact that Mod p, m ,c = Mod p, m ifthe set is compact, give us that Mod p, m is a capacity. For any set L ⊂ M + ( X ) we haveMod p, m ( L ) = sup ε Mod p, m ( L ∩ E ε ), where E ε are the compact sets given by Lemma 3.4.Therefore, suffices to show inner regularity for a Souslin set B contained in E ε for some ε .Since E ε is compact, B is a Souslin-compact set and from Choquet Theorem 2.5 it followsthat for every δ > K ⊂ B such that Mod p, m ( K ) ≥ Mod p, m ( B ) − δ . C p, m is inner regular. Since Souslin sets are universally measurable and M + ( X ) is Polish,we can apply (2.1) to any Souslin set B with σ = η to getsup K ⊂ B C p, m ( K ) = sup K ⊂ B sup c q ( η ) > η ( K ) c q ( η ) = sup c q ( η ) > sup K ⊂ B η ( K ) c q ( η ) = sup c q ( η ) > η ( B ) c q ( η ) = C p, m ( B ) . (cid:3) The duality formula and the existence of maximizers and minimizers provide the fol-lowing result.
Corollary 5.2 (Necessary and sufficient optimality conditions)
Let p > , let Σ ⊂ M + ( X ) be a Souslin set such that Mod p, m (Σ) > and sup Σ µ ( X ) is finite. Then:(a) there exists f ∈ L p + ( X, m ) , unique up to m -negligible sets, such that R X f d µ ≥ for Mod p, m -a.e. µ ∈ Σ and such that k f k pp = Mod p, m (Σ) ;(b) there exists a plan η with barycenter in L q ( X, m ) concentrated on Σ such that C p, m (Σ) =1 /c q ( η ) ;(c) for the function f in (a) and any η in (b) there holds Z X f d µ = 1 for η -a.e. µ and Z X µ d η ( µ ) = f p − k f k pp m . (5.4) Finally, if f ∈ L p + ( X, m ) is optimal in (3.2) , then any plan η with barycenter in L q ( X, m ) concentrated on Σ such that c q ( η ) = k f k − p is optimal in (4.8) . Conversely, if η is optimalin (4.8) , f ∈ L p + ( X, m ) and R X f d µ = 1 for η -a.e. µ then f is optimal in (3.2) . roof. The existence of f follows by Proposition 3.2(iv). The existence of a maximizer η in the duality formula, concentrated on Σ and satisfying C p, m (Σ) = 1 /c q ( η ) follows byLemma 4.4. Since (4.6) gives R X f d µ ≥ η -a.e. µ ∈ Σ we can still derive the inequality(4.5) and obtain from Theorem 5.1 that all inequalities are equalities. Hence, R X f d µ = 1for η -a.e. µ ∈ M + ( X ). Finally, setting µ := R µ d η ( µ ), from (4.3) we get µ = g m with k g k q = c q ( η ). This, in combination with Z X f g d m = Z Z X f dµ d η ( µ ) = c q ( η ) k f k p = k g k q k f k p , gives g = f p − / k f k pp .Finally, the last statements follow directly from (4.5) and Theorem 5.1. (cid:3) In particular, choosing η as in (b) and definingΣ ′ := (cid:26) µ ∈ M + ( X ) : Z X f dµ = 1 (cid:27) , since η (Σ) = η (Σ ′ ) we obtain a subfamily with the same p -modulus on which the constraintis saturated. Part II
Modulus of families of curves and weak gradients
If ( X, d ) is a metric space and I ⊂ R is an interval, we denote by C( I ; X ) the class ofcontinuous maps (often called parametric curves) from I to X . We will use the notation γ t for the value of the map at time t and e t : C( I ; X ) → X for the evaluation map at time t ; occasionally, in order to avoid double subscripts, we will also use the notation γ ( t ). Thesubclass AC( I ; X ) is defined by the property d ( γ s , γ t ) ≤ Z ts g ( r ) d r s, t ∈ I, s ≤ t for some (nonnegative) g ∈ L ( I ). The least, up to L -negligible sets, function g with thisproperty is the so-called metric derivative (or metric speed) | ˙ γ t | := lim h → d ( γ t + h , γ t ) | h | , see [3]. The classes AC p ( I ; X ), 1 ≤ p ≤ ∞ are defined analogously, requiring that | ˙ γ | ∈ L p ( I ). The p -energy of a curve is then defined as E p ( γ ) := (R I | ˙ γ t | p d t if γ ∈ AC p ( I ; X ) , + ∞ otherwise, (6.1)19nd E ( γ ) = ℓ ( γ ), the length of γ , when p = 1. Notice that AC = AC and that AC ∞ ( I ; X )coincides with the class of d -Lipschitz functions.If ( X, d ) is complete the interval I can be taken closed with no loss of generality, becauseabsolutely continuous functions extend continuously to the closure of the interval. Inaddition, if ( X, d ) is complete and separable then C( I ; X ) is a Polish space, and AC p ( I ; X ),1 ≤ p ≤ ∞ are Borel subsets of C( I ; X ) (see for instance [6]). We will use the short notation M + (AC p ( I ; X )) to denote finite Borel measures in C( I ; X ) concentrated on AC p ( I ; X ). We collect in the next proposition a few properties which are well-known in a smoothsetting, but still valid in general metric spaces. We introduce the notationAC ∞ c ([0 , X ) := n σ ∈ AC ∞ ([0 , X ) : | ˙ σ | = ℓ ( σ ) > L -a.e. on (0 , o (6.2)for the subset of AC([0 , X ) consisting of all nonconstant curves with constant speed. Itis easy to check that AC ∞ c ([0 , X ) is a Borel subset of C([0 , X ), since it can also becharacterized by γ ∈ AC ∞ c ([0 , X ) ⇐⇒ < Lip( γ ) ≤ ℓ ( γ ) , (6.3)and the maps γ Lip( γ ) and γ ℓ ( γ ) are lower semicontinuous. Proposition 6.1 (Constant speed reparameterization)
For any γ ∈ AC([0 , X ) with ℓ ( γ ) > , setting s ( t ) := 1 ℓ ( γ ) Z t | ˙ γ r | d r, (6.4) there exists a unique η ∈ AC ∞ c ([0 , X ) such that γ = η ◦ s . Furthermore, η = γ ◦ s − where s − is any right inverse of s . We shall denote by k : (cid:8) γ ∈ AC([0 , X ) : ℓ ( γ ) > (cid:9) → AC ∞ c ([0 , X ) γ η = γ ◦ s − (6.5) the corresponding map.Proof. We prove existence only, the proof of uniqueness being analogous. Les us now definea right inverse, denoted by s − , of s (i.e. s ◦ s − is equal to the identity): we define in theobvious way s − at points y ∈ [0 ,
1] such that s − ( y ) is a singleton; since, by construction, γ is constant in all (maximal) intervals [ c, d ] where s is constant, at points y such that { y } = s ([ c, d ]) we define s − ( y ) by choosing any element of [ c, d ], so that γ ◦ s − ◦ s = γ (even though it could be that s − ◦ s is not the identity). Therefore, if we define η = γ ◦ s − ,we obtain that γ = η ◦ s and that η is independent of the chosen right inverse.In order to prove that η ∈ AC ∞ c ([0 , X ) we define ℓ k := ℓ ( γ )+1 /k and we approximateuniformly in [0 ,
1] the map s by the maps s k ( t ) := ℓ − k R t ( k − + | ˙ γ r | ) d r , whose inverses s − k : [0 , → I are Lipschitz. By Helly’s theorem and passing to the limit as k → ∞ in s k ◦ s − k ( y ) = y , we can assume that a subsequence s − k ( p ) pointwise converges to a right20nverse s − as p → ∞ ; the curves η p := γ ◦ s − k ( p ) are absolutely continuous, pointwiseconverge to η := γ ◦ s − and | η p ( t ) ′ | = | γ ′ ( s − k ( p ) ( t )) | s ′ k ( p ) ( s − k ( p ) ( t )) ≤ ℓ k ( p ) for L -a.e. in t ∈ (0 , η is absolutely continuous and that | ˙ η | ≤ ℓ ( γ ) L -a.e. in (0 , ℓ ( η ) < ℓ ( γ ) providesa contradiction. (cid:3) AC([0 , X ) We can identify curves γ ∈ AC([0 , , X ), ˜ γ ∈ AC([0 , X ) if there exists ϕ : [0 , → [0 , ϕ ∈ AC([0 , , ϕ − ∈ AC([0 , , γ = ˜ γ ◦ ϕ . In thiscase we write γ ∼ ˜ γ . Thanks to the following lemma, the absolute continuity of ϕ − isequivalent to ϕ ′ > L -a.e. in (0 , Lemma 6.2 (Absolute continuity criterion)
Let I , ˜ I be compact intervals in R andlet ϕ : I → ˜ I be an absolutely continuous homeomorphism with ϕ ′ > L -a.e. in I . Then ϕ − : ˜ I → I is absolutely continuous.Proof. Let ψ = ϕ − ; it is a continuous function of bounded variation whose distributionalderivative we shall denote by µ . Since µ ([ a, b ]) = ψ ( b ) − ψ ( a ) for all 0 ≤ a ≤ b ≤
1, weneed to show that µ ≪ L . It is a general property of BV functions (see for instance[4, Proposition 3.92]) that µ ( ψ − ( B )) = 0 for all Borel and L -negligible sets B ⊂ R .Choosing B = ψ ( E ), where E is a L -negligible set where the singular part µ s of µ isconcentrated, the area formula gives Z B ϕ ′ ( s ) d s = L ( E ) = 0 , so that the positivity of ϕ ′ gives L ( B ) = 0. It follows that µ s = 0. (cid:3) Definition 6.3 (The map J ) For any γ ∈ AC([0 , X ) we denote by J γ ∈ M + ( X ) thepush forward under γ of the measure | ˙ γ | L [0 , , namely Z X g d J γ = Z g ( γ t ) | ˙ γ t | d t for all g : X → [0 , ∞ ] Borel. (6.6)
In particular we have that
J γ = J η whenever γ ∼ η and that J γ = J k γ . Although this will not play a role in the sequel, for completeness we provide an intrinsicdescription of the measure
J γ . We denote by H the 1-dimensional Hausdorff measure ofa subset B of X , namely H ( B ) = lim δ ↓ H δ ( B ), where H δ ( B ) := inf ( ∞ X i =0 diam( B i ) : B ⊂ ∞ [ i =0 B i , diam( B i ) < δ ) (with the convention diam( ∅ ) = 0). 21 roposition 6.4 (Area formula) If γ ∈ AC([0 , X ) , then for all g : X → [0 , ∞ ] Borelthe area formula holds: Z g ( γ t ) | ˙ γ t | d t = Z X g ( x ) N ( γ, x ) d H ( x ) , (6.7) where N ( γ, x ) := card( γ − ( x )) is the multiplicity function of γ . Equivalently, J γ = N ( γ, · ) H . (6.8) Proof.
For an elementary proof of (6.7), see for instance [3, Theorem 3.4.6]. (cid:3)
We can now introduce the class of non-parametric curves; notice that we are conventionallyexcluding from this class the constant curves. We introduce the notationAC ([0 , X ) := (cid:8) γ ∈ AC([0 , X ) : | ˙ γ | > L -a.e. on (0 , (cid:9) . It is not difficult to show that AC ([0 , X ) is a Borel subset of C([0 , X ). In addi-tion, Lemma 6.2 shows that for any γ ∈ AC ([0 , X ) the curve k γ ∈ AC ∞ c ([0 , X ) isequivalent to γ . Definition 6.5 (The class C ( X ) of non-parametric curves) The class C ( X ) is de-fined as C ( X ) := AC ([0 , X ) / ∼ , (6.9) endowed with the quotient topology τ C and the canonical projection π C ( X ) . We shall denote the typical element of C ( X ) either by γ or by [ γ ], to mark a distinctionwith the notation used for parametric curves. We will use the notation γ ini and γ fin theinitial and final point of the curve γ ∈ C ( X ), respectively. Definition 6.6 (Canonical maps)
We denote:(a) by i := π C ◦ k : (cid:8) γ ∈ AC([0 , X ) : ℓ ( γ ) > (cid:9) → C ( X ) the projection provided byProposition 6.1, which coincides with the canonical projection π C ( X ) on the quotientwhen restricted to AC ([0 , X ) ;(b) by j := k ◦ π − C : C ( X ) → AC ∞ c ([0 , X ) the canonical representation of a non-parametric curve by a parameterization in [0 , with constant velocity.(c) by ˜ J : C ( X ) → M + ( X ) \ { } the quotient of the map J in (6.6) , defined by ˜ J [ γ ] := J γ. (6.10)22 emma 6.7 (Measurable structure of C ( X ) ) If ( X, d ) is complete and separable, thespace ( C ( X ) , τ C ) is a Lusin Hausdorff space and the restriction of the map i to AC ∞ c ([0 , X ) is a Borel isomorphism. In particular, a collection of curves Γ ⊂ C ( X ) is Borel if andonly if j (Γ) is Borel in C([0 , X ) . Analogously, Γ ⊂ C ( X ) is Souslin if and only if j (Γ) is Souslin in C([0 , X ) .Proof. Let us first show that ( C ( X ) , τ C ) is Hausdorff. We argue by contradiction andwe suppose that there exist curves i ( σ i ) ∈ C ( X ) with σ i ∈ AC ∞ c ([0 , X ), i = 1 ,
2, and asequence of parameterizations s ni ∈ AC([0 , , s ni ) ′ > L -a.e. in (0 , n →∞ sup t ∈ [0 , d ( σ ( s n ( t )) , σ ( s n ( t ))) = 0 . Denoting by r n ( t ) := s n ◦ ( s n ) − and r n ( t ) := s n ◦ ( s n ) − , we getlim n →∞ sup t ∈ [0 , d ( σ ( t ) , σ ( r n ( t ))) = 0 , lim n →∞ sup t ∈ [0 , d ( σ ( r n ( t )) , σ ( t )) = 0 . The lower semicontinuity of the length with respect to uniform convergence yields ℓ := ℓ ( σ ) = ℓ ( σ ) and therefore for every 0 ≤ t ′ < t ′′ ≤ ℓ lim inf n →∞ (cid:0) r n ( t ′′ ) − r n ( t ′ ) (cid:1) = lim n →∞ Z t ′′ t ′ | ( σ ◦ r n ) ′ | d t ≥ Z t ′′ t ′ | σ ′ | d t = ℓ ( t ′′ − t ′ ) . Choosing first t ′ = t and t ′′ = 1 and then t ′ = 0 and t ′′ = t we conclude that lim n r n ( t ) = t for every t ∈ [0 ,
1] and therefore σ = σ .Notice that AC ∞ c ([0 , X ) is a Lusin space, since AC ∞ c ([0 , X ) is a Borel subset ofC([0 , X ). The restriction of i to AC ∞ c ([0 , X ) is thus a continuous and injective mapfrom the Lusin space AC ∞ c ([0 , X ) to the Hausdorff space ( C ( X ) , τ C ) (notice that thetopology τ C is a priori weaker than the one induced by the restriction of i to AC ∞ c ([0 , X )).It follows by definition that C ( X ) is Lusin. Now, Proposition 2.3(iii) yields that therestriction of i is a Borel isomorphism. (cid:3) Lemma 6.8 (Borel regularity of J and ˜ J ) The map J : AC([0 , X ) → M + ( X ) isBorel, where AC([0 , X ) is endowed with the C([0 , X ) topology. In particular, if ( X, d ) is complete and separable the map ˜ J : C ( X ) → M + ( X ) \ { } is Borel and ˜ J (Γ) is Souslinin M + ( X ) whenever Γ is Souslin in C ( X ) .Proof. It is easy to check, using the formula
J γ = γ ♯ ( | ˙ γ | L [0 , J γ = lim n →∞ n − X i =0 d ( γ ( i +1) /n , γ i/n ) δ γ i/n weakly in M + ( X )for all γ ∈ AC([0 , X ) (the simple details are left to the reader). Since the approximatingmaps are continuous, we conclude that J is Borel. The Borel regularity of ˜ J follows byLemma 6.7 and the identity ˜ J = J ◦ j . Since ˜ J is Borel, we can apply Proposition 2.3(iv)to obtain that ˜ J maps Souslin sets to Souslin sets. (cid:3) Modulus of families of non-parametric curves
In this section we assume that ( X, d ) is a complete and separable metric space and that m ∈ M + ( X ).In order to apply the results of the previous sections (with the topology τ inducedby d ) to families of non-parametric curves we consider the canonical map ˜ J : C ( X ) → M + ( X ) \ { } of Definition 6.6(c). In the sequel, for the sake of simplicity, we will notdistinguish between J and ˜ J , writing J γ or J [ γ ] = J γ (this is not a big abuse of notation,since ˜ J is a quotient map).Now we discuss the notion of ( p, m )-modulus, for p ∈ [1 , ∞ ). The ( p, m )-modulus forfamilies Γ ⊂ C ( X ) of non-parametric curves is given byMod p, m (Γ) := inf (Z X g p d m : g ∈ L p + ( X, m ) , Z γ g ≥ γ ∈ Γ ) . (7.1)We adopted the same notation Mod p, m because the identity R γ g = R X g d J γ immediatelygives Mod p, m (Γ) = Mod p, m ( J (Γ)) . (7.2)In a similar vein, setting q = p ′ , in the space C ( X ) we can define plans with barycenterin L q ( X, m ) as Borel probability measures π in C ( X ) satisfying Z C ( X ) J γ d π ( γ ) = g m for some g ∈ L q ( X, m ).Notice that the integral in the left hand side makes sense because the Borel regularity of J easily gives that γ J γ ( A ) is Borel in C ( X ) for all A ∈ B ( X ). We define, exactly asin (4.3), c q ( π ) to be the L q ( X, m ) norm of the barycenter g . Then, the same argumentleading to (4.5) gives π (Γ) c q ( π ) ≤ Mod p, m (Γ) /p for all π ∈ P ( C ( X )) with barycenter in L q ( X, m ) (7.3)for every universally measurable set Γ in C ( X ). Remark 7.1 (Democratic plans)
In more explicit terms, Borel probability measures π in C ( X ) with barycenter in L q ( X, m ) satisfy Z (e t ) ♯ ( | ˙ γ t | π ) d t = g m for some g ∈ L q ( X, m ) (7.4)when we view them as measures on nonconstant curves γ ∈ AC([0 , X ). For instance,in the particular case when π is concentrated on family of geodesics parameterized withconstant speed and with length uniformly bounded from below, the case q = ∞ correspondsto the class of democratic plans considered in [22].24efining C p, m (Γ) as the supremum in the left hand side of (7.3), we can now useTheorem 5.1 to show that even in this case there is no duality gap. Theorem 7.2
For every p > and every Souslin set Γ ⊂ C ( X ) with Mod p, m (Γ) > there exists a π ∈ P (cid:0) C ( X ) (cid:1) with barycenter in L q ( X, m ) , concentrated on Γ and satisfying c q ( π ) = Mod p, m (Γ) − /p .Proof. From Theorem 5.1 we deduce the existence of η ∈ P (cid:0) M + ( X ) (cid:1) with barycenter in L q ( X, m ) concentrated on the Souslin set J (Γ) and satisfying1 c q ( η ) = Mod p, m ( J (Γ)) /p = Mod p, m (Γ) /p . By a measurable selection theorem [11, Theorem 6.9.1] we can find a η -measurable map f : J (Γ) → C ( X ) such that f ( µ ) ∈ Γ ∩ J − ( µ ) for all µ ∈ J (Γ). The measure π := f ♯ η isconcentrated on Γ and the equality between the barycenters Z C ( X ) J γ d π ( γ ) = Z µ d η ( µ )gives c q ( π ) = c q ( η ). (cid:3) In this section we still assume that ( X, d ) is a complete and separable metric space andthat m ∈ M + ( X ). We consider a notion of p -modulus for parametric curves, enforcing thecondition (7.4) (at least when Lipschitz curves are considered), and we compare with thenon-parametric counterpart. To this aim, we introduce the continuous map M : C([0 , X ) → P ( X ) , M ( γ ) := γ ♯ (cid:0) L [0 , (cid:1) . (8.1)Indeed, replacing J γ = γ ♯ ( | ˙ γ | L [0 , M we can consider a “parametric” modulusof a family of curves Σ ⊂ C([0 , X ) just by evaluating Mod p, m ( M (Σ)). By Proposi-tion 3.2(vii), if Σ ⊂ AC ∞ c ([0 , X ) thenMod p, m ( M (Σ)) = 0 ⇐⇒ Mod p, m ( J (Σ)) = 0 . (8.2)On the other hand, things are more subtle when the speed is not constant. Definition 8.1 ( q -energy and parametric barycenter) Let ρ ∈ P (cid:0) C([0 , X ) (cid:1) and q ∈ [1 , ∞ ) . We say that ρ has finite q -energy if ρ is concentrated on AC q ([0 , X ) and Z Z | ˙ γ t | q d t d ρ ( γ ) < ∞ . (8.3) We say that ρ has parametric barycenter h ∈ L q ( X, m ) if Z Z f ( γ t ) d t d ρ ( γ ) = Z X f h d m ∀ f ∈ C b ( X ) . (8.4)25he finiteness condition (8.3) and the concentration on AC q ([0 , X ) can be also bewritten, recalling the definition (6.1) of E q , as follows: Z E q ( γ ) d ρ ( γ ) < ∞ . Notice also that the definition (8.1) of M gives that (8.4) is equivalent to require theexistence of a constant C ≥ Z Z X f d M γ d ρ ( γ ) ≤ C (cid:16) Z X f p d m (cid:17) /p ∀ f ∈ C b ( X ) , f ≥ . (8.5)In this case the best constant C in (8.5) corresponds to k h k L q ( X, m ) for h as in (8.4). Remark 8.2
It is not difficult to check that a Borel probability measure ρ concentratedon a set Γ ⊂ AC ∞ ([0 , X ) with ρ -essentially bounded Lipschitz constants and parametricbarycenter in L q ( X, m ) has also (nonparametric) barycenter in L q ( X, m ). Conversely, if π ∈ P (cid:0) C ( X ) (cid:1) with barycenter in L q ( X, m ) and π -essentially bounded length ℓ ( γ ), then j ♯ π has parametric barycenter in L q ( X, m ).Now, arguing as in the proof of Theorem 7.2 (which provided existence of plans π in C ( X )) we can use a measurable selection theorem to deduce from our basic dualityTheorem 5.1 the following result. Theorem 8.3
For every p > and every Souslin set Σ ⊂ C([0 , X ) , Mod p, m ( M (Σ)) > is equivalent to the existence of ρ ∈ P (cid:0) C([0 , X ) (cid:1) concentrated on Σ with parametricbarycenter in L q ( X, m ) . Our next goal is to use reparameterizations to improve the parametric barycenter from L q ( X, m ) to L ∞ ( X, m ). To this aim, we begin by proving the Borel regularity of someparameterization maps. Let h : X → (0 , ∞ ) be a Borel map with sup X h < ∞ and forevery σ ∈ C([0 , X ) let us set G ( σ ) := Z h ( σ r ) d r, t σ ( s ) := 1 G ( σ ) Z s h ( σ r ) d r : [0 , → [0 , . (8.6)Since t σ is Lipschitz and t ′ σ > L -a.e. in (0 , s σ : [0 , → [0 ,
1] is absolutelycontinuous and we can define H : AC([0 , X ) → AC([0 , X ) , Hσ ( t ) := σ ( s σ ( t )) . (8.7)Notice that H (cid:0) AC ∞ c ([0 , X ) (cid:1) ⊂ AC ([0 , X ). Lemma 8.4 If h : X → R is a bounded Borel function, the map G in (8.6) is Borel. Ifwe assume, in addition, that h > in X , then also t σ in (8.6) is Borel and the map H in (8.7) is Borel and injective. roof. Let us prove first that the map σ ˜ t σ ( t ) = Z t h ( σ r ) d r is Borel from C([0 , X ) to C([0 , h : X → R . Thisfollows by a monotone class argument (see for instance [11, Theorem 2.12.9(iii)]), sinceclass of functions h for which the statement is true is a vector space containing boundedcontinuous functions and stable under equibounded pointwise limits. By the continuity ofthe integral operator, the map G is Borel as well.Now we turn to H , assuming that h >
0. By Proposition 2.3(iii) it will be sufficientto show that the inverse of H , namely the map σ σ ◦ t σ , is Borel. Since the map( σ, t ) σ ◦ t is continuous from C([0 , X ) × C([0 , , X ), the Borel regularityof the inverse of H follows by the Borel regularity of σ t σ . (cid:3) Theorem 8.5
Let q ∈ (1 , ∞ ) and p = q ′ . If ρ ∈ P (cid:0) C([0 , X ) (cid:1) has finite q -energy andparametric barycenter h ∈ L ∞ ( X, m ) , then π = i ♯ ρ has barycenter in L q ( X, m ) and c q ( π ) ≤ (cid:16) Z E q ( γ ) d ρ ( γ ) (cid:17) /q k h k /pL ∞ ( X, m ) . (8.8) Conversely, if π ∈ P (cid:0) C ( X ) (cid:1) has barycenter in L q ( X, m ) and π -essentially bounded length ℓ ( γ ) , concentrated on a Souslin set Γ ⊂ C ( X ) , there exists ρ ∈ P (cid:0) C([0 , X ) (cid:1) with finite q -energy and parametric barycenter in L ∞ ( X, m ) concentrated in a Souslin set containedin [ j (Γ)] .More generally, let σ ∈ P (cid:0) C([0 , X ) (cid:1) be concentrated on a Souslin set Γ ⊂ AC ∞ ([0 , X ) ,with parametric barycenter in L q ( X, m ) and with σ -essentially bounded Lipschitz constants.Then there exists ρ ∈ P (cid:0) C([0 , X ) (cid:1) with finite q -energy and parametric barycenter in L ∞ ( X, m ) concentrated on a Souslin set contained in [Γ] .Proof. Notice that for every nonnegative Borel f there holds Z Z γ f d π ( γ ) = Z Z f ( γ t ) | ˙ γ t | d t d ρ ( γ ) ≤ (cid:16) Z E q d ρ (cid:17) /q (cid:16) Z Z f p ( γ t ) d t d ρ ( γ ) (cid:17) /p ≤ (cid:16) Z E q d ρ (cid:17) /q (cid:16) Z X f p h d m (cid:17) /p ≤ (cid:16) Z E q d ρ (cid:17) /q k h k /pL ∞ ( X, m ) k f k L p ( X, m ) , so that (8.8) holds.Let us now prove the last statement from σ to ρ , since the “converse” statement from π to ρ simply follows by applying the last statement to σ := j ♯ π and recalling Remark 8.2.Let g ∈ L q ( X, m ) be the parametric barycenter of σ and let us set h := 1 / ( ε ∨ g ), with ε > g in a m -negligible set, it is not restrictive to assumethat h is Borel and with values in (0 , /ε ], so that the corresponding maps G and H definedas in (8.6) and (8.7) are Borel. 27e set ˆ ρ := z − G ( · ) σ , where z ∈ (0 , /ε ] is the normalization constant R G ( γ ) d σ ( γ ).Let us consider the inverse s σ : [0 , → [0 ,
1] of the map t σ in (8.6), which is absolutelycontinuous for every σ and the corresponding transformation Hσ in (8.7). We denote by L the σ -essential supremum of the Lipschitz constants of the curves in Γ. Notice that for σ -a.e. σ | ( Hσ ) ′ | ( t ) ≤ L s ′ σ ( t ) = L G ( σ ) h ( Hσ ( t )) L -a.e. in (0 , f one has Z f ( Hσ ( t )) d t = Z f ( σ ( s σ ( t ))) d t = Z f ( σ ( s )) t ′ σ ( s ) d s = 1 G ( σ ) Z f ( σ ( s )) h ( σ ( s )) d s, so that choosing f = h − q and using the inequality G ≤ /ε yields E q ( Hσ ) ≤ L q G q ( σ ) Z h − q ( Hσ ( t )) d t ≤ L q ε q − Z h − q ( σ ( s )) d s. (8.10)Now we set ρ := H ♯ ˆ ρ and notice that, by construction, ρ is concentrated on the Souslinset H (Γ) ⊂ [Γ]. Integrating the q -energy with respect to ρ we obtain Z E q ( θ ) d ρ ( θ ) = Z E q ( Hσ ) dˆ ρ ( σ ) ≤ L q zε q − Z G ( σ ) Z h − q ( σ ( s )) d s d σ ( σ ) ≤ L q zε q Z X gh − q d m = L q zε q Z X g ( ε ∨ g ) q − d m < ∞ , thus obtaining that ρ has finite q -energy. Similarly Z Z f ( θ ( t )) d t d ρ ( θ ) = Z Z f ( Hσ ( t )) d t d ˆ ρ ( σ ) = 1 z Z Z f ( σ ( s )) h ( σ ( s )) d s d σ ( σ )= 1 z Z X f gh d m . Since gh ≤
1, this shows that ρ has parametric barycenter in L ∞ ( X, m ). (cid:3) In the next corollary, in order to avoid further measurability issues, we state our resultwith the inner measure µ ∗ ( E ) := sup { µ ( B ) : B Borel, B ⊂ E } . This formulation is sufficient for our purposes.
Corollary 8.6
A Souslin set Γ ⊂ C ( X ) is Mod p, m -negligible if and only if ρ ∗ ([ j Γ]) = 0 forevery ρ ∈ P (cid:0) C([0 , X ) (cid:1) concentrated on AC q ([0 , X ) and with parametric barycenterin L ∞ ( X, m ) . roof. Let us first suppose that Γ is Mod p, m -negligible and let us denote by h ∈ L ∞ ( X, m )the parametric barycenter of ρ and let us prove that ρ ∗ ([ j Γ]) = 0. Since ρ is concentratedon AC q ([0 , X ) we can assume with no loss of generality (possibly restricting ρ to theclass of curves σ with E q ( σ ) ≤ n and normalizing) that ρ has finite q -energy. We observethat if σ ∈ AC([0 , X ) and f : X → [0 , ∞ ] is Borel, there holds Z f ( σ ( t )) | ˙ σ ( t ) | d t ≤ (cid:16) Z f p ( σ ( t )) d t (cid:17) /p (cid:16) E q ( σ ) (cid:17) /q . (8.11)If f satisfies Z γ f ≥ ∀ γ ∈ Γwe obtain that R σ f ≥ σ ∈ [ j Γ]. We can now integrate w.r.t. ρ and use (8.11) toget ρ ∗ ([ j Γ]) ≤ (cid:16) Z Z f p ( σ ( t )) d t d ρ ( σ ) (cid:17) /p (cid:16) Z E q ( σ ) d ρ ( σ ) (cid:17) /q = (cid:16) Z X f p h d m (cid:17) /p (cid:16) Z E q ( σ ) d ρ ( σ ) (cid:17) /q ≤ k f k p k h k /p ∞ (cid:16) Z E q ( σ ) d ρ ( σ ) (cid:17) /q . (8.12)By minimizing with respect to f we obtain that ρ ∗ ([ j Γ]) = 0.Conversely, suppose that Mod p, m (Γ) >
0; possibly passing to a smaller set, by thecountable subadditivity of Mod p, m we can assume that ℓ is bounded on Γ: then by The-orem 7.2 there exists π ∈ P (cid:0) C ( X ) (cid:1) with barycenter in L q ( X, m ) concentrated on Γ andtherefore the boundedness of ℓ allows to apply the final statement of Theorem 8.5 toobtain ρ ∈ P (cid:0) C([0 , X ) (cid:1) with finite q -energy, parametric barycenter in L ∞ ( X, m ) andconcentrated on a Souslin subset of [ j Γ]. (cid:3)
Corollary 8.7
Let Γ ⊂ AC ∞ ([0 , X ) be a Souslin set such that ρ ∗ (cid:0) [Γ] (cid:1) = 0 for everyplan ρ ∈ P (C([0 , X )) concentrated on AC q ([0 , X ) and with parametric barycenter in L ∞ ( X, m ) . Then M (Γ) is Mod p, m -negligible.Proof. Suppose by contradiction that Mod p, m ( M (Γ)) >
0; possibly passing to a smallerset, by the countable subadditivity of Mod p, m we can assume that Lip is bounded on Γ.By Theorem 8.3 there exists π ∈ P (cid:0) C([0 , X ) (cid:1) with parametric barycenter in L q ( X, m )concentrated on Γ. The boundedness of Lip on Γ allows to apply the second part ofTheorem 8.5 to obtain ρ ∈ P (cid:0) C([0 , X ) (cid:1) with parametric barycenter in L ∞ ( X, m ), finite q -energy and concentrated on a Souslin subset of [Γ]. (cid:3) In this section we will assume that ( X, d ) is a complete and separable metric space and m ∈ M + ( X ) . The following notions have already been used in [6] ( q = 2) and [7] (in29onnection with the Sobolev spaces with gradient in L p ( X, m ), with q = p ′ ; see also [9] inconnection with the BV theory). Definition 9.1 ( q -test plans and negligible sets) Let ρ ∈ P (C([0 , X )) and q ∈ [1 , ∞ ] . We say that ρ is a q -test plan if(i) ρ is concentrated on AC q ([0 , X ) ;(ii) there exists a constant C = C ( ρ ) > satisfying (e t ) ♯ ρ ≤ C m for all t ∈ [0 , .We say that a universally measurable set Γ ⊂ C([0 , X ) is q -negligible if ρ (Γ) = 0 for all q -test plans ρ . Notice that, by definition, C([0 , X ) \ AC q ([0 , X ) is q -negligible. The lack of in-variance of these concepts, even under bi-Lipschitz reparameterizations is due to condition(ii), which is imposed at any given time and with no averaging (and no dependence onspeed as well). Since condition (ii) is more restrictive compared for instance to the notionof democratic test plan of [22] (see Remark 7.1), this means that sets of curves have higherchances of being negligible w.r.t. this notion, as the next elementary example shows.We now want to relate null sets according to Definition 9.1 to null sets in the sense of p -modulus. Notice first that in the definition of q -negligible set we might consider onlyplans ρ satisfying the stronger conditionesssup { E q ( σ ) } < ∞ (9.1)because any q -test plan can be monotonically be approximated by q -test plans satisfyingthis condition. Arguing as in the proof of (8.12) we easily see thatΓ ⊂ C ( X ) Mod p, m -negligible = ⇒ i − (Γ) q -negligible. (9.2)The following simple example shows that the implication can’t be reversed, namely setswhose images under i − are q -negligible need not be Mod p, m -null. Example 9.2
Let X = R , d the Euclidean distance, m = L . The family of parametricsegments Σ = { γ x : x ∈ [0 , } ⊂ AC([0 , R ) with γ xt = ( x, t ) is q -negligible for any q , but i (Σ) has p -modulus equal to 1. In the previous example the implication fails because the trajectories γ x fall, at anygiven time t , into a m -negligible set, and actually the same would be true if this concen-tration property holds at some fixed time. It is tempting to imagine that the implicationis restored if we add to the initial family of curves all their reparameterizations (an opera-tion that leaves the p -modulus invariant). However, since any reparameterization fixes theendpoints, even this fails. However, in the following, we will see that the implicationΓ q -negligible = ⇒ Mod p, m ( i (Γ)) = 030ould be restored if we add some structural assumptions on Γ (in particular a “stability”condition); the collections of curves we are mainly interested in are those connected withthe theory of Sobolev spaces in [6], [7], and we will find a new proof of the fact that if wedefine weak upper gradients according to the two notions, the Sobolev spaces are eventuallythe same.We now fix some additional notation: for I = [ a, b ] ⊂ [0 ,
1] we define the “stretching”map s I : AC([0 , X ) → AC([0 , X ), mapping γ to γ ◦ s I , where s I : [0 , → [ a, b ] is theaffine map with s I (0) = a and s I (1) = b . Notice that this map acts also in all the otherspaces AC q , AC , AC ∞ c of parametric curves we are considering. Definition 9.3 (Stable and invariant sets of curves) (i) We say that Γ ⊂ { γ ∈ AC([0 , X ) : ℓ ( γ ) > } is invariant under constant speedreparameterization if k γ ∈ Γ for all γ ∈ Γ ;(ii) We say that Γ ⊂ AC([0 , X ) is ∼ -invariant if [ γ ] ⊂ Γ for all γ ∈ Γ ;(iii) We say that Γ ⊂ AC([0 , X ) is stable if for every γ ∈ Γ there exists ε ∈ (0 , / such that s I γ ∈ Γ whenever I = [ a, b ] ⊂ [0 , and | a | + | − b | ≤ ε . The following theorem provides key connections between q -negligibility and Mod p, m -negligibility, both in the nonparametric sense (statement (i)) and in the parametric case(statement (ii)), for stable sets of curves. Theorem 9.4
Let Γ ⊂ AC([0 , X ) be a Souslin and stable set of curves.(i) If, in addition, ℓ ( γ ) > for all γ ∈ Γ and Γ is both ∼ -invariant and invariantunder constant speed reparameterization, then Γ is q -negligible if and only if J (Γ) is Mod p, m -negligible in M + ( X ) (equivalently, i (Γ) is Mod p, m -negligible in C ( X ) ).(ii) If Γ is q -negligible and [Γ ∩ AC ∞ ([0 , X )] ⊂ Γ , then M (cid:0) Γ ∩ AC ∞ ([0 , X ) (cid:1) is Mod p, m -negligible in M + ( X ) . If Γ is also ∼ -invariant then the converse holds, too.Proof. (i) The proof of the nontrivial implication, from positivity of Mod p, m ( J (Γ)) to Γbeing not q -negligible is completely analogous to the proof of (ii), given below, by applyingCorollary 8.6 to i (Γ) in place of Corollary 8.7 to Γ ∩ AC ∞ ([0 , X ) and the same rescalingtechnique. Since we will only need (ii) in the sequel, we only give a detailed proof of (ii).(ii) Let us prove that the positivity of Mod p, m (cid:0) M (Γ ∩ AC ∞ ([0 , X )) (cid:1) implies thatΓ is not q -negligible. Since Γ ∩ AC ∞ ([0 , X ) is stable, we can assume the existence of ε ∈ (0 , /
2) such that s I γ ∈ Γ whenever I = [ a, b ] ⊂ [0 ,
1] and | a | + | − b | ≤ ε .By applying Corollary 8.7 to Γ ∩ AC ∞ ([0 , X ) we obtain the existence of ρ ∈ P (cid:0) AC q ([0 , X ) (cid:1) concentrated on a Souslin subset of [Γ ∩ AC ∞ ([0 , X )], and then on Γ, with L ∞ parametricbarycenter, i.e. such that Z (e t ) ♯ ρ d t ≤ C m for some C > . (9.3)31et’s define a family of reparameterization maps F τε : AC q ([0 , X ) → AC q ([0 , X ): F τε γ ( t ) = γ (cid:16) t + τ ε (cid:17) t ∈ [0 , , ∀ γ ∈ AC q ([0 , X ) , ∀ τ ∈ [0 , ε ] . (9.4)Let us consider now the measure ρ ε = 1 ε Z ε ( F τε ) ♯ ρ dτ. We claim that ρ ε is a q -plan: it is clear that ρ ε is a probability measure on AC q ([0 , X ),and so we have to check only the marginals at every time:(e t ) ♯ ρ ε = 1 ε Z ε (e t ) ♯ (cid:0) ( F τε ) ♯ ρ (cid:1) dτ = 1 ε Z ε (e t + τ ε ) ♯ ρ dτ = 1 + εε Z t + ε εt ε (e s ) ♯ ρ ds ≤ εε Z (e s ) ♯ ρ ds ≤ C εε m for all t ∈ [0 , . Now we reach the absurd if we show that ρ ε is concentrated on Γ; in order to do so it issufficient to notice that F τε = s I with I = I τε = [ τ ε , τ ε ] and τ ∈ [0 , ε ].Now, if we assume also that [Γ] ⊂ Γ, then we know that for all γ ∈ Γ the curve η := γ ◦ s − belongs to Γ ∩ AC ∞ ([0 , X ), where s : [0 , → [0 ,
1] is the parameterization definedin the proof of Proposition 6.1. We recall that by definition we have (1+ ℓ ( γ )) s ′ ( t ) = 1+ | ˙ γ t | for L -a.e. t ; in particular, the change of variables formula gives Z (1 + | ˙ γ t | ) g ( γ t ) d t = (1 + ℓ ( γ )) Z g ( η s ) d s ∀ g : X → [0 , ∞ ] Borel. (9.5)We suppose that M (cid:0) Γ ∩ AC ∞ ([0 , X ) (cid:1) is Mod p, m -negligible; this gives us f ∈ L p + ( X, m )such that Z f ( η s ) d s = ∞ ∀ η ∈ Γ ∩ AC ∞ ([0 , X ) . (9.6)Now given any q -plan π we have that Z Γ Z ( | ˙ γ t | + 1) f ( γ t ) d t d π ( γ ) ≤ (cid:18)Z Z ( | ˙ γ t | + 1) q d t d π ( γ ) (cid:19) /q (cid:18)Z Z f p ( γ t )d t d π ( γ ) (cid:19) /p ≤ (cid:18)(cid:16)Z E q d π (cid:17) /q + 1 (cid:19) (cid:18) C ( π ) · Z X f p d m (cid:19) /p < ∞ (9.7)Now, using (9.6) and (9.5) with g = f give R ( | ˙ γ t | + 1) f ( γ t ) d t = ∞ for all γ ∈ Γ, so that(9.7) gives that π (Γ) = 0. Since π is arbitrary, Γ is q -negligible. (cid:3) Remark 9.5
We note that the proof shows that if Γ is ∼ -invariant and M (cid:0) Γ ∩ AC ∞ ([0 , X ) (cid:1) is Mod p, m -negligible in M + ( X ), then Γ is q -negligible, independently of the stability as-sumption that we used in the converse implication.32 As in the previous sections, ( X, d ) will be a complete and separable metric space and m ∈ M + ( X ) . Recall that a Borel function g : X → [0 , ∞ ] is an upper gradient of f : X → R if | f ( γ fin ) − f ( γ ini ) | ≤ Z γ g (10.1)holds for all γ ∈ C ( X ). Here, the curvilinear integral R γ g is given by R J g ( γ t ) | ˙ γ t | d t , where γ : J → X is any parameterization of the curve γ (i.e., γ = i γ , and one can canonicallytake γ = j γ ). It follows from Proposition 6.4 that the upper gradient property can beequivalently written in the form | f ( γ fin ) − f ( γ ini ) | ≤ Z X g d J γ.
Now we introduce two different notions of Sobolev function and a corresponding notion of p -weak gradient; the first one was first given in [23] while the second one in [6] for p = 2and in [7] for general exponent. When discussing the corresponding notions of (minimal)weak gradient we will follow the terminology of [7]. Definition 10.1 ( N ,p and p -upper gradient) Let f be a m -measurable and p -integrablefunction on X . We say that f belongs to the space N ,p ( X, d , m ) if there exists g ∈ L p + ( X, m ) such that (10.1) is satisfied for Mod p, m -a.e. curve γ . Functions in N ,p have the important Beppo-Levi property of being absolutely contin-uous along Mod p, m -a.e. curve γ (more precisely, this means f ◦ j γ ∈ AC([0 , N ,p ( X, d , m ) belong to theSobolev space defined below (see [6], [7]) where (10.1) is required for q -a.e. curve γ . Definition 10.2 ( W ,p and p -weak upper gradient) Let f be a m -measurable and p -integrable function on X . We say that f belongs to the space W ,p ( X, d , m ) if there exists g ∈ L p + ( X, m ) such that | f ( γ ) − f ( γ ) | ≤ Z g ( γ t ) | ˙ γ t | d t is satisfied for q -a.e. curve γ ∈ AC q ([0 , X ) . We remark that there is an important difference between the two definitions, namelythe first one is a priori not invariant if we change the function f on a m -negligible set, whilethe second one has this kind of invariance, because for any q -test plan ρ , any m -negligibleBorel set N and any t ∈ [0 ,
1] the set { γ : γ t ∈ N } is ρ -negligible. Associated to these twonotions are the minimal p -upper gradient and the minimal p -weak upper gradient, bothuniquely determined up to m -negligible sets (for a more detailed discussion, see [7, 23]).33s an application of Theorem 9.4, we show that these two notions are essentially equiv-alent modulo the choice of a representative in the equivalence class: more precisely, forany f ∈ W ,p ( X, d , m ) there exists a m -measurable representative ˜ f of f which belongs to N ,p ( X, d , m ). This result is not new, because in [6] and [7] the equivalence has alreadybeen shown. On the other hand, the proof of the equivalence in [6] and [7] is by no meanselementary, it passes through the use of tools from the theory of gradient flows and optimaltransport theory and it provides the equivalence with another relevant notion of “relaxed”gradient based on the approximation through Lipschitz functions. We provide a totallydifferent proof, using the results proved in this paper about negligibility of sets of curves.In the following theorem we provide, first, existence of a “good representative” of f .Notice that the standard theory of Sobolev spaces provides existence of this representativevia approximation with Lipschitz functions. Theorem 10.3 (Good representative)
Let f : X → R be a Borel function and let usset Γ = (cid:8) γ ∈ AC ∞ ([0 , X ) : f ◦ γ has a continuous representative f γ : [0 , → R (cid:9) . If Mod p, m (cid:0) M (AC ∞ ([0 , X ) \ Γ) (cid:1) = 0 there exists a m -measurable representative ˜ f : X → R of f satisfying Mod p, m (cid:0) M ( { γ ∈ Γ : ˜ f ◦ γ f γ } ) (cid:1) = 0 . (10.2) In particular(i) for q -a.e. curve γ there holds ˜ f ◦ γ ≡ f γ ;(ii) for Mod p, m -a.e. curve γ there holds ˜ f ◦ j γ ≡ f j γ .Proof. Let us set ˜Γ := AC ∞ ([0 , X ) \ Γ, so that our assumption reads Mod p, m ( M (˜Γ)) = 0.Notice first that the (ii) makes sense because f j γ exists for Mod p, m -a.e. curve γ thanks to(8.2) and Mod p, m ( M (˜Γ ∩ AC ∞ c ([0 , X ))) = 0 (also, constant curves are all contained inΓ). Also, (i) makes sense thanks to Remark 9.5 and to the fact that the defining propertyof Γ is ∼ -invariant.Step 1. (Construction of a good set Γ g of curves). Since we have Mod p, m ( M (˜Γ)) = 0, thereexists h ∈ L p + ( X, m ) such that R h ◦ σ = ∞ for every σ ∈ ˜Γ. Starting from Γ and h , wecan define the set Γ g = (cid:8) η ∈ Γ : R h ◦ η < ∞ (cid:9) of “good” curves, satisfying the followingthree conditions:(a) f ◦ η has a continuous representative for all η ∈ Γ g ;(b) R h ◦ η < ∞ for all η ∈ Γ g ;(c) M (cid:0) AC ∞ ([0 , X ) \ Γ g (cid:1) is Mod p, m -negligible.34ndeed, properties (a) and (b) follow easily by definition, while (c) follows by the inclusion M (cid:0) AC ∞ ([0 , X ) \ Γ g (cid:1) ⊂ M (cid:0) AC ∞ ([0 , X ) \ Γ (cid:1) ∪ (cid:8) µ : Z X h d µ = ∞ (cid:9) . Step 2. (Construction of ˜ f ). For every point x ∈ X we consider the set of pairs goodcurves-times that pass through x at time t :Θ x = { ( η, t ) ∈ Γ g × [0 ,
1] : η ( t ) = x } , and, thanks to property (a) of Γ g , we can partition this set according to the value of thecontinuous representative f η at t :Θ x = [ r ∈ R Θ rx with Θ rx = { ( η, t ) ∈ Θ x : f η ( t ) = r } . Now, the key point is that for every x ∈ X there exists at most one r such Θ rx is not empty.Indeed, suppose that r = r and that there exist ( η , t ) ∈ Θ r x , ( η , t ) ∈ Θ r x , so that r = f η ( t ) = f η ( t ) = r ; since η , η ∈ Γ g , property (b) of Γ g gives Z h ◦ η d t + Z h ◦ η d t < ∞ . (10.3)Suppose to fix the ideas that t > t < η ∈ AC ∞ ([0 , X ) byconcatenation: η ( s ) := ( η (2 st ) if s ∈ [0 , / ,η (1 − − s )(1 − t )) if s ∈ [1 / , . This curve is clearly absolutely continuous and it follows first η for half of the time andthen it follows η ; it is clear that, since f ◦ η coincides L -a.e. in (0 ,
1) with the function a ( s ) := ( f η (2 st ) if s ∈ [0 , / ,f η (1 − − s )(1 − t )) if s ∈ [1 / , s = 1 / f ◦ η has no continuous representative. Itfollows that η belongs to ˜Γ and therefore R h ◦ η = ∞ . But, since12 t Z h ◦ η d t + 12(1 − t ) Z h ◦ η d t ≥ Z h ◦ η d t we get a contradiction with (10.3).Now we define˜ f ( x ) := ( f η ( t ) if ( η, t ) ∈ Θ x for some η ∈ Γ g , t ∈ [0 , f ( x ) otherwise.35y construction, ˜ f ( η ( t )) = f η ( t ) for all t ∈ [0 ,
1] and η ∈ Γ g , so that property (c) of Γ g shows (10.2). Using Remark 9.5 and the fact that { γ ∈ AC([0 , X ) : ˜ f ◦ γ ≡ f γ } isclearly ∼ -invariant, we obtain (i) from (10.2). Moreover, from (10.2) we get in particularthat Mod p, m (cid:0) M ( { γ ∈ Γ ∩ AC ∞ c ([0 , X ) : ˜ f ◦ γ f γ } ) (cid:1) = 0 . (10.4)Recalling (8.2) and the fact that j is a Borel isomorphism, we can rewrite (10.4) asMod p, m (cid:0) J ( { γ ∈ C ( X ) : ˜ f ◦ j γ f j γ } ) (cid:1) = 0 , and so we proved also (ii).Step 3. (The set F := { f = ˜ f } is m -negligible.) Let γ x be the curve identically equal x , that is γ xt = x for all t ∈ [0 , γ x belongs to Γ for every x ∈ X : inparticular f γ x ( t ) = f ( x ) for every t ∈ [0 , c of constant curves γ x satisfying ˜ f ◦ γ x f γ x , then f ( x ) = ˜ f ( x ) for every such curve,hence ˜Γ c = { γ x : x ∈ F } . In particular we have that M (˜Γ c ) = { δ x : x ∈ F } . Now, from(10.2), we know that Mod p, m ( M (˜Γ c )) = 0; this provides the existence of g ∈ L p + ( X, m ) suchthat g ( x ) = ∞ for every x ∈ F , and so we get that F is contained in a m -negligible set. (cid:3) The following simple example shows that, in Theorem 10.3, the “nonparametric” assump-tion that J (AC([0 , X ) \ Γ) is Mod p, m -negligible is not sufficient to conclude that ˜ f = f m -a.e. in X . Example 10.4
Let X = [0 , d the Euclidean distance, m = L + δ / , p ∈ [1 , ∞ ). Thefunction f identically equal to 0 on X \ { / } and equal to 1 at x = 1 / f j γ for Mod p, m -a.e. curve γ , but any function˜ f such that ˜ f ◦ j γ ≡ f j γ for Mod p, m -a.e. γ should be equal to 0 also at x = 1 /
2, so that m ( { f = ˜ f } ) = 1.Now, we are going to apply Theorem 10.3 to the problem of equivalence of Sobolevspaces. We begin with a few preliminary results and definitions.Let f : X → R , g : X → [0 , ∞ ] be Borel functions. We consider the sets I ( g ) := n γ ∈ AC([0 , X ) : Z γ g < ∞ o , (10.5)and B ( f, g ) := n γ ∈ I ( g ) : f ◦ γ ∈ W , (0 , , | dd t ( f ◦ γ ) | ≤ | ˙ γ | g ◦ γ L -a.e. in (0 , o . (10.6)We will need the following simple measure theoretic lemma, which says that integra-tion in one variable maps Borel functions to Borel functions. Its proof is an elementaryconsequence of a monotone class argument (see for instance [11, Theorem 2.12.9(iii)]) andof the fact that the statement is true for F bounded and continuous.36 emma 10.5 Let ( Y, d Y ) be a metric space and let F : [0 , × Y → [0 , ∞ ] be Borel. Thenthe function I F : Y → [0 , ∞ ] defined by y R F ( t, y ) d t is a Borel function. Lemma 10.6
Let f : X → R , g : X → [0 , ∞ ] be Borel functions. Then I ( g ) \ B ( f, g ) is aBorel set, stable and ∼ -invariant.Proof. Stability is simple to check: if, by contradiction, it were γ ∈ I ( g ) \ B ( f, g )and s [ a n ,b n ] γ ∈ B ( f, g ) with a n ↓ b n ↑
1, we would get f ◦ γ ∈ W , ( a n , b n ) and | dd t f ◦ γ | ≤ | ˙ γ | g ◦ γ ∈ L (0 , L -a.e. in ( a n , b n ). Taking limits, we would obtain γ ∈ B ( f, g ),a contradiction.For the proof of ∼ -invariance we note that, first of all, that Lemma 10.5 with F ( t, γ ) := g ( γ t ) | ˙ γ t | guarantees that I ( g ) is a ∼ -invariant Borel set, provided we define F using a Borelrepresentative of | ˙ γ | ; this can be achieved, for instance, using the lim inf of the metricdifference quotients. Analogously, the set L := (cid:8) γ ∈ AC([0 , X ) : Z | f ( γ t ) | d t < ∞ (cid:9) is Borel. Now, γ ∈ B ( f, g ) if and only if γ ∈ I ( g ) ∩ L and (cid:12)(cid:12)(cid:12)(cid:12)Z φ ′ ( t ) f ( γ t ) d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | φ ( t ) | g ( γ t ) | ˙ γ t | d t for all φ ∈ W (10.7)with W = { φ ∈ AC([0 , , φ (0) = φ (1) = 0 } . Now, if both s and s − are absolutelycontinuous from [0 ,
1] to [0 , η := γ ◦ s , we can use the change of variables formulato obtain that ( φ ◦ s ) ′ f ◦ η ∈ L (0 ,
1) for all φ ∈ W and that (cid:12)(cid:12)(cid:12)(cid:12)Z ( φ ◦ s ) ′ ( r ) f ( η r ) d r (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | φ ◦ s ( r ) | g ( η r ) | ˙ η r | d r for all φ ∈ W. Since W ◦ s = W we eventually obtain φ ′ f ◦ η ∈ L (0 ,
1) for all φ ∈ W (so that f ◦ η islocally integrable in (0 , (cid:12)(cid:12)(cid:12)(cid:12)Z φ ′ ( r ) f ( η r ) d r (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | φ ( r ) | g ( η r ) | ˙ η r | d r for all φ ∈ W. (10.8)It is easy to check that these two conditions, in combination with R η g < ∞ , imply that η ∈ L , therefore f ◦ η belongs to B ( f, g ) and ∼ -invariance is proved.In order to prove that B ( f, g ) is Borel we follow a similar path: we already know thatboth I ( g ) and L are Borel, and then in the class I ( g ) ∩ L the condition (10.8), now with W replaced by a countable dense subset of C c (0 ,
1) for the C norm, provides a characterizationof B ( f, g ). Since for φ ∈ C c (0 ,
1) fixed the maps η ∈ L Z φ ′ ( r ) f ( η r ) d r, η Z | φ ( r ) | g ( η r ) | ˙ η r | d r are easily seen to be Borel in AC([0 , X ) (as a consequence of Lemma 10.5, splitting inpositive and negative part the first integral and using once more a Borel representative of | ˙ η | in the second integral) we obtain that B ( f, g ) is Borel. (cid:3) heorem 10.7 (Equivalence theorem) Any f ∈ N ,p ( X, d , m ) belongs to W ,p ( X, d , m ) .Conversely, for any f ∈ W ,p ( X, d , m ) there exists a m -measurable representative ˜ f thatbelongs to N ,p ( X, d , m ) . More precisely, ˜ f satisfies:(i) ˜ f ◦ γ ∈ AC([0 , for q -a.e. curve γ ∈ AC([0 , X ) ;(ii) ˜ f ◦ j γ ∈ AC([0 , for Mod p, m -a.e. curve γ .Proof. We already discussed the easy implication from N ,p to W ,p , so let us focus onthe converse one. In the sequel we fix f ∈ W ,p ( X, d , m ) and a p -weak upper gradient g . By Fubini’s theorem, it is easily seen that the space W ,p ( X, d , m ) is invariant undermodifications in m -negligible sets; as a consequence, since the Borel σ -algebra is countablygenerated, we can assume with no loss of generality that f is Borel. Another simpleapplication of Fubini’s theorem (see [7, Remark 4.10]) shows that for q -a.e. curve γ thereexists an absolutely continuous function f γ : [0 , → R such that f γ = f ◦ γ L -a.e. in(0 ,
1) and | dd t f γ | ≤ | ˙ γ | g ◦ γ L -a.e. in (0 , L p integrability of g yields that thecomplement of I ( g ) is q -negligible, we can use Lemma 10.6 and Theorem 9.4(ii) to inferthat Σ = I ( g ) \ B ( f, g ) satisfies Mod p, m (cid:0) M (Σ ∩ AC ∞ ([0 , X )) (cid:1) = 0.By Theorem 10.3 we obtain a m -measurable representative ˜ f of f such that ˜ f ◦ γ ≡ f γ for q -a.e. curve γ and ˜ f ◦ j γ ≡ f j γ for Mod p, m -a.e. γ . Hence, the fundamental theorem ofcalculus for absolutely continuous functions gives | ˜ f ( γ fin ) − ˜ f ( γ ini ) | = | f j γ (1) − f j γ (0) | ≤ Z g (( j γ ) t ) | ˙( j γ ) t | d t = Z γ g for Mod p, m -a.e. γ . (cid:3) References [1]
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