Null-finite sets in metric groups and their applications
aa r X i v : . [ m a t h . GN ] N ov NULL-FINITE SETS IN TOPOLOGICAL GROUPS AND THEIR APPLICATIONS
TARAS BANAKH AND ELIZA JAB LO ´NSKA
Abstract.
In the paper we introduce and study a new family of “small” sets which is tightly connected withtwo well known σ -ideals: of Haar-null sets and of Haar-meager sets. We define a subset A of a topologicalgroup X to be null-finite if there exists a convergent sequence ( x n ) n ∈ ω in X such that for every x ∈ X the set { n ∈ ω : x + x n ∈ A } is finite. We prove that each null-finite Borel set in a complete metric Abelian group isHaar-null and Haar-meager. The Borel restriction in the above result is essential as each non-discrete metricAbelian group is the union of two null-finite sets. Applying null-finite sets to the theory of functional equationsand inequalities, we prove that a mid-point convex function f : G → R defined on an open convex subset G ofa metric linear space X is continuous if it is upper bounded on a subset B which is not null-finite and whoseclosure is contained in G . This gives an alternative short proof of a known generalization of Bernstein-Doetschtheorem (saying that a mid-point convex function f : G → R defined on an open convex subset G of a metriclinear space X is continuous if it is upper bounded on a non-empty open subset B of G ). Since Borel Haar-finitesets are Haar-meager and Haar-null, we conclude that a mid-point convex function f : G → R defined on anopen convex subset G of a complete linear metric space X is continuous if it is upper bounded on a Borelsubset B ⊂ G which is not Haar-null or not Haar-meager in X . The last result resolves an old problem in thetheory of functional equations and inequalities posed by Baron and Ger in 1983. Introduction
In 1920 Steinhaus [29] proved that for any measurable sets
A, B of positive Haar measure in a locally compactPolish group X the sum A + B := { a + b : a ∈ A, b ∈ B } has non-empty interior in X and the difference B − B := { a − b : a, b ∈ B } is a neighborhood of zero in X .In [8] Christensen extended the “difference” part of the Steinhaus results to all Polish Abelian groups provingthat for a Borel subset B of a Polish Abelian group X the difference B − B is a neighborhood of zero if B isnot Haar-null. Christensen defined a Borel subset B ⊂ X to be Haar-null if there exists a Borel σ -additiveprobability measure µ on X such that µ ( B + x ) = 0 for all x ∈ X .A topological version of Steinhaus theorem was obtained by Pettis [25] and Piccard [26] who proved that forany non-meager Borel sets A, B in a Polish group X the sum A + B has non-empty interior and the difference B − B is a neighborhood of zero.In 2013 Darji [9] introduced a subideal of the σ -ideal of meager sets in a Polish group, which is a topologicalanalog to the σ -ideal of Haar-null sets. Darji defined a Borel subset B of a Polish group X to be Haar-meager ifthere exists a continuous map f : K → X from a non-empty compact metric space K such that for every x ∈ X the preimage f − ( B + x ) is meager in K . By [18], for every Borel subset B ⊂ X which is not Haar-meager ina Polish Abelian group X the difference B − B is a neighborhood of zero.It should be mentioned that in contrast to the “difference” part of the Steinhaus and Piccard–Pettis theorems,the “additive” part can not be generalized to non-locally compact groups: by [23], [18] each non-locally compactPolish Abelian group X contains two closed sets A, B whose sum A + B has empty interior but A, B are neitherHaar-null nor Haar-meager in X . In the Polish group X = R ω , for such sets A, B we can take the positivecone R ω + .Steinhaus-type theorems has significant applications in the theory of functional equations and inequalities.For example, the “additive” part of Steinhaus Theorem can be applied to prove that a mid-point convexfunction f : R n → R is continuous if it is upper bounded on some measurable set B ⊂ R n of positive Lebesguemeasure (see e.g. [20, p.210]). We recall that a function f : G → R defined on a convex subset of a linear spaceis mid-point convex if f (cid:0) x + y (cid:1) ≤ f ( x )+ f ( y )2 for any x, y ∈ G . Unfortunately, due to the example of Matou˘skov´aand Zelen´y [23] we know that the “additive” part of the Steinhaus theorem does not extend to non-locally Mathematics Subject Classification.
Key words and phrases.
Automatic continuity, additive function, mid-point convex function, null-finite set, Haar-null set,Haar-meager set. compact Polish Abelian groups. This leads to the following natural problem whose “Haar-null” version wasposed by Baron and Ger in [3, P239].
Problem (Baron, Ger, 1983).
Is the continuity of an additive or mid-point convex function f : X → R on aBanach space X equivalent to the upper boundedness of the function on some Borel subset B ⊂ X which is notHaar-null or not Haar-meager? In this paper we give the affirmative answer to the Baron–Ger Problem applying a new concept of a null-finite set, which will be introduced in Section 2. In Section 3 we show that non-null-finite sets A in metricgroups X possess a Steinhaus-like property: if a subset A of a metric group X is not null-finite, then A − ¯ A is a neighborhood of θ and for some finite set F ⊂ X the set F + ( A − A ) is a neighborhood of zero in X .In Sections 5 and 6 we shall prove that Borel null-finite sets in Polish Abelian groups are Haar-meager andHaar-null. On the other hand, in Example 7.1 we show that the product Q n ∈ N C n of cyclic groups contains aclosed Haar-null set which is not null-finite and in Example 7.2 we show that the product Q n ∈ N C n contains anull-finite G δ -set which cannot be covered by countably many closed Haar-null sets. In Section 8 we prove thateach non-discrete metric Abelian group is the union of two null-finite sets and also observe that sparse sets(in the sense of Lutsenko–Protasov [22]) are null-finite. In Section 9 we apply null-finite sets to prove that anadditive functional f : X → R on a Polish Abelian group is continuous if it is upper bounded on some subset B ⊂ X which is not null-finite in X . In Section 10 we generalize this result to additive functions with valuesin Banach and locally convex spaces. Finally, in Section 11, we prove a generalization of Bernstein–Doetschtheorem and next apply it in the proof of a continuity criterion for mid-point convex functions on completemetric linear spaces, thus answering the Baron–Ger Problem. In Section 12 we pose some open problemsrelated to null-finite sets. 1. Preliminaries
All groups considered in this paper are Abelian.
The neutral element of a group will be denoted by θ . For agroup G by G ∗ we denote the set of non-zero elements in G . By C n = { z ∈ C : z n = 1 } we denote the cyclicgroup of order n .By a ( complete ) metric group we understand an Abelian group X endowed with a (complete) invariantmetric k · − · k . The invariant metric k · − · k determines (and can be recovered from) the prenorm k · k definedby k x k := k x − θ k . So, a metric group can be equivalently defined as a group endowed with a prenorm.Formally, a prenorm on a group X is a function k · k : X → R + := [0 , ∞ ) satisfying three axioms: • k x k = 0 iff x = θ ; • k − x k = k x k ; • k x + y k ≤ k x k + k y k for any x, y ∈ X ; see [1, § R of real numbers. By a metric linear space weunderstand a linear space endowed with an invariant metric.A non-empty family I of subsets of a set X is called ideal of sets on X if I satisfies the following conditions: • X / ∈ I ; • for any subsets J ⊂ I ⊂ X the inclusion I ∈ I implies J ∈ I ; • for any sets A, B ∈ I we have A ∪ B ∈ I .An ideal I on X is called a σ -ideal if for any countable subfamily C ⊂ I the union S C belongs to X .An ideal I on a group X is called invariant if for any I ∈ I and x ∈ X the shift x + I of I belongs to theideal I .For example, the family [ X ] <ω of finite subsets of an infinite group X is an invariant ideal on X .A topological space is Polish if it is homeomorphic to a separable complete metric space. A topologicalspace is analytic if it is a continuous image of a Polish space.2.
Introducing null-finite sets in topological groups
In this section we introduce the principal new notion of this paper.A sequence ( x n ) n ∈ ω in a topological group X is called a null-sequence if it converges to the neutral element θ of X . ULL-FINITE SETS IN TOPOLOGICAL GROUPS AND THEIR APPLICATIONS 3
Definition 2.1.
A set A of a topological group X is called null-finite if there exists a null-sequence ( x n ) n ∈ ω in X such that for every x ∈ X the set { n ∈ ω : x + x n ∈ A } is finite.Null-finite sets in metrizable topological groups can be defined as follows. Proposition 2.2.
For a non-empty subset A of a metric group X the following conditions are equivalent: (1) A is null-finite; (2) there exists an infinite compact set K ⊂ X such that for every x ∈ X the intersection K ∩ ( x + A ) isfinite; (3) there exists a continuous map f : K → X from an infinite compact space K such that for every x ∈ X the preimage f − ( A + x ) is finite.Proof. The implications (1) ⇒ (2) ⇒ (3) are obvious, so it is enough to prove (3) ⇒ (1). Assume that f : K → X is a continuous map from an infinite compact space K such that for every x ∈ X the preimage f − ( A + x ) is finite. Fix any point a ∈ A . It follows that for every x ∈ X the fiber f − ( a + x ) ⊂ f − ( A + x ) isfinite and hence the image f ( K ) is infinite. So, we can choose a sequence ( y n ) n ∈ ω of pairwise distinct pointsin f ( K ). Because of the compactness and metrizability of f ( K ), we can additionally assume that the sequence( y n ) n ∈ ω converges to some point y ∞ ∈ f ( K ). Then the sequence ( x n ) n ∈ ω of points x n := y n − y ∞ , n ∈ ω , is anull-sequence witnessing that the set A is null-finite. (cid:3) Null-finite sets are “small” in the following sense.
Proposition 2.3.
For any null-finite set A in a metric group X and any finite subset F ⊂ X the set F + A has empty interior in X .Proof. Since A is null-finite, there exists a null-sequence ( x n ) n ∈ ω in X such that for every x ∈ X the set { n ∈ ω : x + x n ∈ A } is finite.To derive a contradiction, assume that for some finite set F the set F + A has non-empty interior U in X .Then for any point u ∈ U the set { n ∈ ω : u + x n ∈ U } ⊂ { n ∈ ω : u + x n ∈ F + A } is infinite. By the PigeonholePrinciple, for some y ∈ F the set { n ∈ ω : u + x n ∈ y + A } is infinite and so is the set { n ∈ ω : − y + u + x n ∈ A } .But this contradicts the choice of the sequence ( x n ) n ∈ ω . (cid:3) Now we present some examples of closed sets which are (or are not) null-finite. We recall that for a group G by G ∗ we denote the set G \ { θ } of non-zero elements of G . Example 2.4.
Let ( G n ) n ∈ ω be a sequence of finite groups. The set A := Q n ∈ ω G ∗ n is null-finite in the compactmetrizable topological group G = Q n ∈ ω G n if and only if lim n →∞ | G n | 6 = ∞ .Proof. First, we assume that lim n →∞ | G n | = ∞ and show that the set A is not null-finite in G . Given anull-sequence ( x n ) n ∈ ω in the compact topological group G , we should find an element a ∈ G such that the set { n ∈ ω : a + x n ∈ A } is infinite.It will be convenient to think of elements of the group G as functions x : ω → L n ∈ ω G n such that x ( k ) ∈ G k for all k ∈ ω .Taking into account that lim k →∞ | G k | = ∞ and lim n →∞ x n = θ ∈ G , we can inductively construct anincreasing number sequence ( n k ) k ∈ ω such that n = 0 and for every k ∈ N and m ≥ n k the following twoconditions are satisfied:(1) x m ( i ) = θ for all i < n k − ;(2) | G m | ≥ k + 3.We claim that for every i ≥ n the set { x n k ( i ) : k ∈ ω } has cardinality < | G i | . Indeed, given any i ≥ n , wecan find a unique j ≥ n j ≤ i < n j +1 and conclude that x n k ( i ) = θ for all k ≥ j + 1 (by the condition(1)). Then the set { x n k ( i ) : k ∈ ω } = { θ } ∪ { x n k ( i ) : k ≤ j } has cardinality ≤ j + 1) < j + 3 ≤ | G i | . Thelast inequality follows from i ≥ n k and condition (2). Therefore, the set { x n k ( i ) : k ∈ ω } has cardinality < | G i | and we can choose a point a i ∈ G i \ {− x n k ( i ) : k ∈ ω } and conclude that a i + x n k ( i ) ∈ G ∗ i for all k ∈ ω . Thenthe element a = ( a i ) i ∈ ω ∈ G has the required property as the set { n ∈ ω : a + x n ∈ A } ⊃ { n k } k ∈ ω is infinite.Next, assuming that lim k →∞ | G k | 6 = ∞ , we shall prove that the set A is Haar-finite in G . Since lim k →∞ | G k | 6 = ∞ , for some l ∈ ω the set Λ = { k ∈ ω : | G k | = l } is infinite.Since lim n →∞ (1 − l ) n = 0, there exists n ∈ N such that (1 − l ) n < l and hence l ( l − n < l n . This impliesthat for every k ∈ Λ the set G k · ( G ∗ k ) n := { ( x + a , · · · x + a n ) : x ∈ G k , a , . . . , a n ∈ G ∗ k } has cardinality | G k · ( G ∗ k ) n | ≤ l ( l − n < l n = | G nk | . Consequently, the (compact) set G · A n = { ( x + a , . . . , x + a n ) : x ∈ TARAS BANAKH AND ELIZA JAB LO´NSKA
G, a , . . . , a n ∈ A } is nowhere dense in the compact topological group G n . Now Theorem 2.5 ensures that A is null-finite in G . (cid:3) Theorem 2.5.
A non-empty subset A of a Polish group X is null-finite in X if for some n ∈ N the set X · A n := { ( x + a , . . . , x + a n ) : x ∈ X, a , . . . , a n ∈ A } is meager in X n .Proof. Let K ( X ) be the space of non-empty compact subsets of X , endowed with the Vietoris topology. Fora compact subset K ⊂ X let K n ∗ = { ( x , . . . , x n ) ∈ K n : |{ x , . . . , x n }| = n } the set of n -tuples consisting ofpairwise distinct points of K .By Mycielski-Kuratowski Theorem [19, 19.1], the set W ∗ ( A ) := { K ∈ K ( X ) : K n ∗ ∩ ( X · A n ) = ∅} is comeagerin K ( X ) and hence contains some infinite compact set K . We claim that for every x ∈ X the set K ∩ ( x + A )has cardinality < n . Assuming the opposite, we could find n pairwise distinct points a , . . . , a n ∈ A such that x + a i ∈ K for all i ≤ n . Then ( x + a , . . . , x + a n ) ∈ K n ∗ ∩ X · A n , which contradicts the inclusion K ∈ W ∗ ( A ).By Proposition 2.2, the set A is null-finite. (cid:3) A Steinhaus-like properties of sets which are not null-finite
In [29] Steinhaus proved that for a subset A of positive Lebesgue measure in the real line, the set A − A is a neighborhood of zero. In this section we establish three Steinhaus-like properties of sets which are notnull-finite. Theorem 3.1.
If a subset A of a metric group X is not null-finite, then (1) the set A − ¯ A is a neighborhood of θ in X ; (2) each neighborhood U ⊂ X of θ contains a finite subset F ⊂ U such that F + ( A − A ) is a neighborhoodof θ . (3) If X is Polish (and A is analytic), then A − A is not meager in X (and ( A − A ) − ( A − A ) is aneighborhood of θ in X ).Proof.
1. Assuming that A − ¯ A is not a neighborhood of θ , we could find a null-sequence ( x n ) n ∈ ω containedin X \ ( A − ¯ A ). Since A is not null-finite, there exists a ∈ X such that the set Ω = { n ∈ ω : a + x n ∈ A } isinfinite. Then a ∈ { a + x n } n ∈ Ω ⊂ ¯ A and hence x n = ( a + x n ) − a ∈ A − ¯ A for all n ∈ Ω, which contradicts thechoice of the sequence ( x n ) n ∈ ω .2. Fix a decreasing neighborhood base ( U n ) n ∈ ω at zero in X such that U ⊂ U . For the proof by con-tradiction, suppose that for any finite set F ⊂ U the set F + ( A − A ) is not a neighborhood of zero. Thenwe can inductively construct a null-sequence ( x n ) n ∈ ω such that x n ∈ U n \ S ≤ i Let f : C ω → C be a discontinuous homomorphism on the compact Polish group X = C ω (thought as the vector space C ω over the two-element field Z / Z ). Then the subgroup A = f − ( θ ) is notnull-finite in X (by Proposition 2.3), but A − A = A = ( A − A ) − ( A − A ) is not a neighborhood of θ in X .4. A combinatorial characterization of null-finite sets in compact metric groups In compact metric groups null-finite sets admit a purely combinatorial description. Proposition 4.1. A non-empty subset A of a compact metric group X is null-finite if and only if there existsan infinite set I ⊂ X such that for any infinite subset J ⊂ I the intersection T x ∈ J ( A − x ) is empty. ULL-FINITE SETS IN TOPOLOGICAL GROUPS AND THEIR APPLICATIONS 5 Proof. To prove the “only if” part, assume that A ⊂ X is null-finite. So, there exists a null-sequence ( x n ) n ∈ ω such that for every x ∈ X the set { n ∈ ω : x + x n ∈ A } is finite. It follows that for every x ∈ X the set { n ∈ ω : x n = x } is finite and hence the set I := { x n } n ∈ ω is infinite. We claim that this set has the requiredproperty. Indeed, assuming that for some infinite subset J ⊂ I the intersection T x ∈ J ( A − x ) contains somepoint a ∈ X , we conclude that the set { n ∈ ω : a + x n ∈ A } contains the set { n ∈ ω : x n ∈ J } and hence isinfinite, which contradicts the choice of the sequence ( x n ) n ∈ ω .To prove the “if” part, assume that there exists an infinite set I ⊂ X such that for every infinite set J ⊂ I the intersection T x ∈ J ( A − x ) is empty. By the compactness of the metric group X , some sequence( x n ) n ∈ ω of pairwise distinct points of the infinite set I converges to some point x ∞ ∈ X . Then the null-sequence ( z n ) n ∈ ω consisting of the points z n = x n − x ∞ , n ∈ ω , witnesses that A is null-finite. Assumingthe opposite, we would find a point a ∈ X such that the set { n ∈ ω : a + z n ∈ A } is infinite. Then the set J := { x ∈ I : a − x ∞ + x ∈ A } ⊂ { x n : n ∈ ω, a − x ∞ + x n ∈ A } is infinite, too, and the intersection T x ∈ J ( A − x ) contains the point a − x ∞ and hence is not empty, which contradicts the choice of the set I . (cid:3) Following Lutsenko and Protasov [22] we define a subset A of an infinite group X to be sparse if for anyinfinite set I ⊂ X there exists a finite set F ⊂ I such that T x ∈ F ( x + A ) is empty. By [22, Lemma 1.2] thefamily of sparse subsets of a group is an invariant ideal on X (in contrast to the family of null-finite sets). Proposition 4.2. Each sparse subset A of a non-discrete metric group X is null-finite.Proof. Being non-discrete, the metric group X contains a null-sequence ( x n ) n ∈ ω consisting of pairwise distinctpoints. Assuming that the sparse set A is not null-finite, we can find a point a ∈ X such that the setΩ := { n ∈ ω : a + x n ∈ A } is infinite. Since A is sparse, for the infinite set I := {− x n : n ∈ Ω } there exists afinite set F ⊂ I such that the intersection T x ∈ F ( x + A ) ⊃ T m ∈ Ω ( − x n + A ) ∋ a is empty, which is not possibleas this intersection contains the point a . (cid:3) Null-finite Borel sets are Haar-meager In this section we prove that each null-finite set with the universal Baire property in a complete metricgroup is Haar-meager.A subset A of a topological group X is defined to have the universal Baire Property (briefly, A is a uBP-set )if for any function f : K → X from a compact metrizable space K the preimage f − ( A ) has the Baire Propertyin K , which means that for some open set U ⊂ K the symmetric difference U △ f − ( A ) is meager in K . It iswell-known that each Borel subset of a topological group has the universal Baire Property.A uBP-set A of a topological group X is called Haar-meager if there exists a continuous map f : K → X from a compact metrizable space K such that f − ( x + A ) is meager in K for all x ∈ X . By [9], for a completemetric group X the family HM X of subsets of Haar-meager uBP-sets in X is an invariant σ -ideal on X . Formore information on Haar-meager sets, see [9], [11], [10], [18]. Theorem 5.1. Each null-finite uBP-set in a compete metric group is Haar-meager.Proof. To derive a contradiction, suppose that a null-finite uBP-set A in a complete metric group ( X, k · k ) isnot Haar-meager. Since A is null-finite, there exists a null-sequence ( a n ) n ∈ ω such that for every x ∈ X the set { n ∈ ω : x + a n ∈ A } is finite.Replacing ( a n ) n ∈ ω by a suitable subsequence, we can assume that k a n k < n for all n ∈ ω . For every n ∈ ω consider the compact set K n := { θ } ∪ { a m } m ≥ n ⊂ X . The metric restriction on the sequence ( a n ) n ∈ ω impliesthat the function Σ : Y n ∈ ω K n → X, Σ : ( x n ) n ∈ ω X n ∈ ω x n , is well-defined and continuous (the proof of this fact can be found in [18]; see the proof of Theorem 2).Since the set A is not Haar-meager and has uBP, there exists a point z ∈ X such that B := Σ − ( z + A )is not meager and has the Baire Property in the compact metrizable space K := Q n ∈ ω K n . Consequently,there exits a non-empty open set U ⊂ K such that U ∩ B is comeager in U . Replacing U by a smaller subset,we can assume that U is of basic form { b } × Q m ≥ j K m for some j ∈ ω and some element b ∈ Q m In this section we prove that each universally measurable null-finite set in a complete metric group is Haar-null.A subset A of a topological group X is defined to be universally measurable if A is measurable with respect toany σ -additive Radon Borel probability measure on X . It is clear that each Radon Borel subset of a topologicalgroup is universally measurable. We recall that a measure µ on a topological space X is Radon if for any ε > K ⊂ X such that µ ( X \ K ) < ε .A universally measurable subset A of a topological group X is called Haar-null if there exists a σ -additiveRadon Borel probability measure µ on X such that µ ( x + A ) = 0 for all x ∈ X . By [8], for any complete metricgroup X the family HN X of subsets of universally measurable Haar-null subsets of X is an invariant σ -ideal in X . For more information on Haar-null sets, see [8], [16], [17] and also e.g. [13], [15], [23], [28]. Theorem 6.1. Each universally measurable null-finite set A in a complete metric group X is Haar-null.Proof. To derive a contradiction, suppose that the null-finite set A is not Haar-null in X . Since A is null-finite,there exists a null-sequence ( a n ) n ∈ ω such that for each x ∈ X the set { n ∈ ω : x + a n ∈ A } is finite. Replacing( a n ) n ∈ ω by a suitable subsequence, we can assume that k a n k ≤ n for all n ∈ ω .Consider the compact space Π := Q n ∈ ω { , , . . . , n } endowed with the product measure λ of uniformlydistributed measures on the finite discrete spaces { , , , , . . . , n } , n ∈ ω . Consider the mapΣ : Π → X, Σ : ( p i ) i ∈ ω ∞ X i =0 p i a i . Since k p i a i k ≤ i k a i k ≤ i i = i , the series P ∞ i =0 p i a i is convergent and the function Σ is well-defined andcontinuous.Since the set A is not Haar-null, there exists an element z ∈ X such that the preimage Σ − ( z + A ) haspositive λ -measure and hence contains a compact subset K of positive measure. For every n ∈ ω considerthe subcube Π n := Q n − i =0 { , , , . . . , i } × Q ∞ i = n { , , . . . , i } of Π and observe that λ (Π n ) → n → ∞ .Replacing K by K ∩ Π l for a sufficiently large l , we can assume that K ⊂ Π l . For every m ≥ l let s m : Π l → Πbe the “back-shift” defined by the formula s m (( x i ) i ∈ ω ) := ( y i ) i ∈ ω where y i = x i for i = m and y i = x i − i = m . Claim 6.2. For any compact set C ⊂ Π l of positive measure λ ( C ) and any ε > there exists k ≥ l such thatfor any m ≥ k the intersection C ∩ s m ( C ) has measure λ ( C ∩ s m ( C )) > (1 − ε ) λ ( C ) .Proof. By the regularity of the measure λ , the set C has a neighborhood O ( C ) ⊂ Π such that λ ( O ( C ) \ C ) <ελ ( K ). By the compactness of C , there exists k ≥ l such that for any m ≥ k the shift s m ( C ) is contained in O ( C ). Hence λ ( s m ( C ) \ C ) ≤ λ ( O ( C ) \ C ) < ελ ( C ) and thus λ ( s m ( C ) ∩ C ) = λ ( s m ( C )) − λ ( s m ( C ) \ C ) >λ ( C ) − ελ ( C ) = (1 − ε ) λ ( C ). (cid:3) Using Claim 6.2 we can choose an increasing number sequence ( m k ) k ∈ ω such that m > l and the set K ∞ := T k ∈ ω s m k ( K ) has positive measure and hence contains a point ~b := ( b i ) i ∈ ω . It follows that for every k ∈ ω the point ~b k := s − m k ( ~b ) belongs to K ⊂ Σ − ( z + A ).Observe that Σ( ~b k ) = Σ( ~b ) + a m k ∈ z + A and hence − z + Σ( ~b ) + a m k ∈ A for all k ∈ ω , which contradictsthe choice of the sequence ( a n ) n ∈ ω . (cid:3) Remark 6.3. After writing the initial version of this paper we discovered that a result similar to Theorem 6.1was independently found by Bingham and Ostaszewski [6, Theorem 3].7. The σ -ideal generated by (closed) Borel null-finite sets In this section we introduce two new invariant σ -ideals generated by (closed) Borel null-finite sets and studythe relation of these new ideals to the σ -ideals of Haar-null and Haar-meager sets.Namely, for a complete metric group X let • σ NF X be the smallest σ -ideal containing all Borel null-finite sets in X ; ULL-FINITE SETS IN TOPOLOGICAL GROUPS AND THEIR APPLICATIONS 7 • σ NF X be the smallest σ -ideal containing all closed null-finite sets in X ; • σ HN X be the smallest σ -ideal containing all closed Haar-null sets in X .Theorems 5.1 and 6.1 imply that σ NF X ⊂ σ HN X and σ NF X ⊂ HN X ∩ HM X . So, we obtain the followingdiagram in which an arrow A → B indicates that A ⊂ B . σ NF X / / (cid:15) (cid:15) σ HN X (cid:15) (cid:15) σ NF X / / HN X ∩ HM X In Examples 7.1 and 7.2 we show that the σ -ideal σ NF X is strictly smaller than σ HN X and the ideal σ NF X isnot contained in σ HN X . Example 7.1. The closed set A = Q n ≥ C ∗ n in the product X = Q n ≥ C n of cyclic groups is Haar-null butcannot be covered by countably many closed null-finite sets. Consequently, A ∈ σ HN X \ σ NF X .Proof. The set A = Q ∞ n =2 C ∗ n has Haar measure ∞ Y n =2 | C ∗ n || C n | = ∞ Y n =2 n − n = 0and hence is Haar-null in the compact Polish group X .Next, we show that A cannot be covered by countably many closed null-finite sets. To derive a contradiction,assume that A = S n ∈ ω A n where each A n is closed and null-finite in X . By the Baire Theorem, for some n ∈ ω the set A n has non-empty interior in A . Consequently, we can find m > a ∈ Q m − n =2 C ∗ n such that { a } × Q ∞ n = m C ∗ n ⊂ A n . By Example 2.4, the set Q ∞ n = m C ∗ n is not null-finite in Q ∞ n = m C n , which implies thatthe set A n ⊃ { a } × Q ∞ n = m C ∗ n is not null-finite in the group X . But this contradicts the choice of A n . (cid:3) Our next example shows that σ NF X σ HN X for some compact Polish group X . Example 7.2. For any function f : ω → [2 , ∞ ) with Q n ∈ ω f ( n ) − f ( n ) > , the compact metrizable group X = Q n ∈ N C f ( n ) contains a null-finite G δ -set A ⊂ X which cannot be covered by countably many closed Haar-nullsets in X . Consequently, A ∈ NF X \ σ HN X .Proof. For every n ∈ ω let g n be a generator of the cyclic group C f ( n ) . In the compact metrizable group X = Q n ∈ ω C f ( n ) consider the null-sequence ( x n ) n ∈ ω defined by the formula x n ( i ) = ( θ if i ≤ ng i if i > n. Consider the closed subset B = Q n ∈ ω C ∗ f ( n ) in X and observe that it has positive Haar measure, equal to theinfinite product Y n ∈ ω f ( n ) − f ( n ) > . It is easy to see that for every n ∈ ω the set C ∗ f ( n ) = C f ( n ) \ { θ } is not equal to C ∗ f ( n ) ∩ ( g n + C ∗ f ( n ) ), whichimplies that the intersection B ∩ ( x n + B ) is nowhere dense in B . Consequently, the set A = B \ S n ∈ ω ( x n + B )is a dense G δ -set in B .We claim that the set A is null-finite. Given any a ∈ X we should prove that the set { n ∈ ω : a + x n ∈ A } isfinite. If a / ∈ B , then the open set X \ B is a neighborhood of a in X . Since the sequence ( a + x n ) n ∈ ω convergesto a ∈ X \ B , the set { n ∈ ω : a + x n ∈ B } ⊃ { n ∈ ω : a + x n ∈ A } is finite.If a ∈ B , then { n ∈ ω : a + x n ∈ A } ⊂ { n ∈ ω : ( B + x n ) ∩ A = ∅} = ∅ by the definition of the set A .Next, we prove that the G δ -set A cannot be covered by countably many closed Haar-null sets. To derive acontradiction, assume that A ⊂ S n ∈ ω F n for some closed Haar-null sets F n ⊂ X . Since the space A is Polish,we can apply the Baire Theorem and find n ∈ ω such that the set A ∩ F n has non-empty interior in A andhence its closure A ∩ F n has non-empty interior in B = Q n ∈ ω C ∗ f ( n ) . It is easy to see that each non-emptyopen subset of B has positive Haar measure in X . Consequently, the set A ∩ F n has positive Haar measure,which is not possible as this set is contained in the Haar-null set F n . (cid:3) TARAS BANAKH AND ELIZA JAB LO´NSKA Remark 7.3. Answering a question posed in a preceding version of this paper, Adam Kwela [21] constructedtwo compact null-finite subsets A, B on the real line, whose union A ∪ B is not null-finite. This means thatthe family of subsets of (closed) Borel null-finite subsets on the real line is not an ideal, and the ideal σ NF R contains compact subsets of the real line, which fail to be null-finite.8. Decomposing non-discrete metric groups into unions of two null-finite sets By Theorem 6.1 and the countable additivity of the family of Borel Haar-null sets [8], the countable unionof Borel null-finite sets in a complete metric group X is Haar-null in X and hence is not equal to X . So, acomplete metric group cannot be covered by countably many Borel null-finite sets. This result dramaticallyfails for non-Borel null-finite sets. Theorem 8.1. Each non-discrete metric group X can be written as the union X = A ∪ B of two null-finitesubsets A, B of X .Proof. Being non-discrete, the metric group X contains a non-trivial null-sequence, which generates a non-discrete countable subgroup Z in X .Let Z = { z n } n ∈ ω be an enumeration of the countable infinite group Z such that z = θ and z n = z m forany distinct numbers n, m ∈ ω . By induction we can construct sequences ( u n ) n ∈ ω and ( v n ) n ∈ ω in Z such that u = v = θ and for every n ∈ N the following two conditions are satisfied:(1) k u n k ≤ n and u n / ∈ {− z i + z j + v k : i, j ≤ n, k < n } ∪ {− z i + z j + u k : i, j ≤ n, k < n } ;(2) k v n k ≤ n and v n / ∈ {− z i + z j + u k : i, j ≤ n, k ≤ n } ∪ {− z i + z j + v k : i, j ≤ n, k < n } .At the n -th step of the inductive construction, the choice of the points u n , v n is always possible as the ball { z ∈ Z : k z k ≤ n } is infinite and u n , v n should avoid finite sets.After completing the inductive construction, we obtain the null-sequences ( u n ) n ∈ ω and ( v n ) n ∈ ω of pairwisedistinct points of Z such that for any points x, y ∈ Z the intersection { x + u n } n ∈ ω ∩ { y + u m } m ∈ ω is finite andfor every distinct points x, y ∈ Z the intersections { x + u n } n ∈ ω ∩ { y + u m } m ∈ ω and { x + v n } n ∈ ω ∩ { y + v m } m ∈ ω are finite.Using these facts, for every n ∈ ω we can choose a number i n ∈ ω such that the set { z n + v k : k ≥ i n } isdisjoint with the set { z i + v m : i < n, m ∈ ω } ∪ { z i + u m : i ≤ n, m ∈ ω } and the set { z n + u k : k ≥ i n } isdisjoint with the set { z i + u m : i < n, m ∈ ω } ∪ { z i + v m : i ≤ n, m ∈ ω } .We claim that the set A := S n ∈ ω { z n + u m : m > i n } is null-finite in Z . This will follow as soon as we verifythe condition:(3) for any z ∈ Z the set { n ∈ ω : z + v n ∈ A } is finite.Find j ∈ ω such that z = z j and observe that for every n ≥ j the choice of the number i n guarantees that { z n + u m } m>i n ∩ { z j + v n } n ∈ ω = ∅ , so { n ∈ ω : z + v n ∈ A } is contained in the finite set S n Corollary 8.2. Let X be a non-discrete metric group. ULL-FINITE SETS IN TOPOLOGICAL GROUPS AND THEIR APPLICATIONS 9 (1) Each null-finite subset of X is contained in some invariant ideal on X ; (2) For any ideal I on X there exists a null-finite set A ⊂ X such that A / ∈ I .Proof. 1. For any null-finite set A in X the family I A = { I ⊂ X : ∃ F ∈ [ X ] <ω I ⊂ F A } is an invariant ideal on X whose elements have empty interiors in X by Proposition 2.3. We recall that by[ X ] <ω we denote the ideal of finite subsets of X .2. By Theorem 8.1, the group X can be written as the union X = A ∪ B of two null-finite sets. Then forany ideal I on X one of the sets A or B does not belong to I . (cid:3) Applying null-finite sets to additive functionals In this section we apply null-finite sets to prove a criterion of continuity of additive functionals on metricgroups.A function f : X → Y between groups is called additive if f ( x + y ) = f ( x ) + f ( y ) for every x, y ∈ X . Anadditive function into the real line is called an additive functional . Theorem 9.1. An additive functional f : X → R on a metric group X is continuous if it is upper bounded ona set B ⊂ X which is not null-finite.Proof. Suppose that the functional f is not continuous. Then there exists ε > f ( U ) ( − ε, ε )for each neighborhood U ⊂ X of zero. It follows that for every n ∈ ω there is a point x n ∈ X such that k x n k ≤ n and | f ( x n ) | > ε . Observe that, k n x n k ≤ n · k x n k ≤ n and | f (2 n x n ) | = 2 n | f ( x n ) | > n ε . Choose ε n ∈ { , − } such that ε n f (2 n x n ) is positive and put z n := ε n n x n . Then f ( z n ) > n ε and k z n k ≤ n .We claim that the null-sequence ( z n ) n ∈ ω witnesses that the set B is null-finite. Given any point x ∈ X weneed to check that the set Ω := { n ∈ ω : x + z n ∈ B } is finite.Let M := sup f ( B ) and observe that for every n ∈ Ω we have M ≥ f ( x + z n ) = f ( x ) + f ( z n ) ≥ f ( x ) + 2 n ε, which implies 2 n ≤ ε ( M − f ( x )). Hence the set Ω is finite. Therefore, the null-sequence ( z n ) n ∈ ω witnessesthat the set B is null-finite, which contradicts our assumption. (cid:3) Theorems 9.1 and 5.1 imply Corollary 9.2. An additive functional f : X → R on a complete metric group is continuous if and only if itis upper bounded on some uBP set B ⊂ X which is not Haar-meager. An analogous result for Haar-null sets follows from Theorems 9.1 and 6.1. Corollary 9.3. An additive functional f : X → R on a complete metric group is continuous if and only if itis upper bounded on some universally measurable set B ⊂ X which is not Haar-null. Applying null-finite sets to additive functions In this section we prove some continuity criteria for additive functions with values in Banach spaces orlocally convex spaces. Corollary 10.1. An additive function f : X → Y from a complete metric group X to a Banach space Y iscontinuous if for any linear continuous functional y ∗ : Y → R the function y ∗ ◦ f : X → R is upper boundedon some set B ⊂ X which is not null-finite in X .Proof. Assuming that f is discontinuous, we can find ε > U ⊂ X of θ we get f ( U ) B ε := { y ∈ Y : k y k < ε } . Then for every n ∈ ω we can find a point x n ∈ X such that k x n k ≤ n and k f ( x n ) k ≥ ε . Since k n x n k ≤ n k x n k ≤ n , the sequence (2 n x n ) n ∈ ω is a null-sequence in X .On the other hand, k f (2 n x n ) k = 2 n k f ( x n ) k ≥ n ε for all n ∈ ω , which implies that the set { f (2 n x n ) } n ∈ ω isunbounded in the Banach space Y . By the Banach-Steinhaus Uniform Boundedness Principle [14, 3.15], thereexists a linear continuous functional y ∗ : Y → R such that the set { y ∗ ◦ f (2 n x n ) : n ∈ ω } is unbounded in R . By our assumption, the additive functional y ∗ ◦ f : X → R is upper bounded on some set B ⊂ X whichis not null-finite in X . By Theorem 9.1, the additive functional y ∗ ◦ f is continuous and thus the sequence (cid:0) y ∗ ◦ f (2 n x n ) (cid:1) n ∈ ω converges to zero and hence cannot be unbounded in R . This is a desired contradiction,completing the proof. (cid:3) Corollary 10.1 admits a self-generalization. Theorem 10.2. An additive function f : X → Y from a complete metric group X to a locally convex space Y is continuous if for any linear continuous functional y ∗ : Y → R the function y ∗ ◦ f : X → R is upper boundedon some set B ⊂ X which is not null-finite in X .Proof. This theorem follows from Corollary 10.1 and the well-known fact [27, p.54] that each locally convexspace is topologically isomorphic to a linear subspace of a Tychonoff product of Banach spaces. (cid:3) Combining Theorem 10.2 with Theorems 5.1 and 6.1, we derive two corollaries. Corollary 10.3. An additive function f : X → Y from a complete metric group X to a locally convex space Y is continuous if for any linear continuous functional y ∗ : Y → R the function y ∗ ◦ f : X → R is upper boundedon some uBP-set B ⊂ X which is not Haar-meager in X . Corollary 10.4. An additive function f : X → Y from a complete metric group X to a locally convex space Y is continuous if for any linear continuous functional y ∗ : Y → R the function y ∗ ◦ f : X → R is upper boundedon some universally measurable set B ⊂ X which is not Haar-null in X . Applying null-finite sets to mid-point convex functions In this section we apply null-finite sets to establish a continuity criterion for mid-point convex functions onlinear metric spaces.A function f : C → R on a convex subset C of a linear space is called mid-point convex if f ( x + y ) ≤ f ( x ) + f ( y )for any x, y ∈ C . Mid-point convex functions are alternatively called Jensen convex . Theorem 11.1. A mid-point convex function f : G → R defined on an open convex subset G ⊂ X of a metriclinear space X is continuous if and only if f is upper bounded on some set B ⊂ G which is not null-finite in X and whose closure B is contained in G .Proof. The “only if” part is trivial. To prove the “if” part, assume that f is upper bounded on some set B ⊂ G such that B ⊂ G and B is not null-finite in X . We need to check the continuity of f at any point c ∈ G .Shifting the set G and the function f by − c , we may assume that c = θ . Also we can replace the function f by f − f ( θ ) and assume that f ( θ ) = 0. In this case the mid-point convexity of f implies that f ( n x ) ≤ n f ( x )for every x ∈ G and n ∈ N .To derive a contradiction, suppose that the function f is not continuous at θ . Then there exists ε > f ( U ) ( − ε, ε ) for any open neighborhood U ⊂ G . For every n ∈ ω consider the neighborhood U n := { x ∈ G : k x k ≤ n , x ∈ ( n G ) ∩ ( − n G ) } and find a point x n ∈ U n such that | f ( x n ) | ≥ ε . Replacing x n by − x n ,if necessary, we can assume that f ( x n ) ≥ ε (this follows from the inequality ( f ( x n ) + f ( − x n )) ≥ f ( θ ) = 0ensured by the mid-point convexity of f ).Since x n ∈ n G , for every k ≤ n the point 2 k x n belongs to G and has prenorm k k x n k ≤ k k x n k ≤ n n = n in the metric linear space ( X, k · k ). This implies that (2 n x n ) n ∈ ω is a null-sequence in X . Since the set B isnot null-finite, there exists a ∈ X such that the set Ω := { n ∈ N : a + 2 n x n ∈ B } is infinite. Then a ∈ B ⊂ G .Choose k ∈ N so large that 2 − k +1 a ∈ − G (such number k exists as − G is a neighborhood of θ ).Observe that for every number n ∈ Ω with n > k the mid-point convexity of f ensures that2 n − k ε ≤ n − k f ( x n ) ≤ f (2 n − k x n ) = f (cid:0) − − k +1 a + − k +1 a +2 n − k +1 x n (cid:1) ≤≤ f ( − − k +1 a ) + f (2 − k +1 a + 2 n − k +1 x n ) ≤ f ( − − k +1 a ) + 2 − k f ( a + 2 n x n )and hence sup f ( B ) ≥ sup n ∈ Ω f ( a + 2 n x n ) ≥ sup k Corollary 11.2. A mid-point-convex function f : G → R defined on an open convex subset G of a metriclinear space X is continuous if and only if f is upper-bounded on some non-empty open subset of G . ULL-FINITE SETS IN TOPOLOGICAL GROUPS AND THEIR APPLICATIONS 11 Combining Theorem 11.1 with Theorems 5.1 and 6.1, we obtain the following two continuity criteria, whichanswer the Problem of Baron and Ger [3, P239]. Corollary 11.3. A mid-point convex function f : G → R defined on a convex subset G of a complete metriclinear space X is continuous if and only if it is upper bounded on some uBP-set B ⊂ G which is not Haar-meagerin X .Proof. Assume that f is upper bounded on some uBP-set B ⊂ G which is not Haar-meager in X .Write the open set G of X as the union U = S n ∈ ω F n of closed subsets of X . By [9], the countable union ofuBP Haar-meager sets is Haar-meager. Since the set B = S n ∈ ω B ∩ F n is not Haar-meager, for some n ∈ N thesubset B ∩ F n of B is not Haar-meager. By Theorem 5.1, B ∩ F n is not null-finite. Since B ∩ F n ⊂ F n ⊂ G and f is upper bounded on B ∩ F n , we can apply Theorem 11.1 and conclude that the mid-point convex function f is continuous. (cid:3) Corollary 11.4. A mid-point convex function f : G → R defined on a convex subset G of a complete metriclinear space X is continuous if and only if it is upper bounded on some universally measurable set B ⊂ G whichis not Haar-null in X .Proof. The proof of this corollary runs in exactly the same way as the proof of Corollary 11.3 and uses thewell-known fact [8] that the countable union of universally measurable Haar-null sets in a complete metricgroup is Haar-null. (cid:3) Some Open Problems In this section we collect some open problems related to null-finite sets.It is well-known that for a locally compact Polish group X , each Haar-meager set in X can be enlargedto a Haar-meager F σ -set and each Haar-null set in X can be enlarged to a Haar-null G δ -set in X . Those“enlargement” results dramatically fail for non-locally compact Polish groups, see [13], [10]. We do not knowwhat happens with null-finite sets in this respect. Problem 12.1. Is each Borel null-finite subset A of a (compact) Polish group X contained in a null-finite set B ⊂ X of low Borel complexity? By [28] (resp. [12, 4.1.6]), each analytic Haar-null (resp. Haar-meager) set in a Polish group is containedin a Borel Haar-null (resp. Haar-meager) set. On the other hand, each non-locally compact Polish groupcontains a coanalytic Haar-null (resp. Haar-meager) set which cannot be enlarged to a Borel Haar-null (resp.Haar-meager) set, see [13] (resp. [10]). Problem 12.2. Is each (co)analytic null-finite set A in a Polish group X contained in a Borel null-finite set? Our next problem ask about the relation of the σ -ideal σ NF X generated by null-finite sets to other known σ -ideals. Problem 12.3. Let X be a Polish group. (1) Is σ HN X ⊂ σ NF X ? (2) Is σ NF X = HN X ∩ HM X ? The negative answer to both parts of Problem 12.3 would follow from the negative answer to the followingconcrete question. Problem 12.4. 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