Generalization and New Proof for Almost Everywhere Convergence to Imply Local Convergence in Measure
aa r X i v : . [ m a t h . G M ] A p r Generalization and New Proof for Almost EverywhereConvergence to Imply Local Convergence in Measure
Yu-Lin Chou ∗ Abstract
With a new proof approach we prove in a more general setting the classical convergencetheorem that almost everywhere convergence of measurable functions on a finite measurespace implies convergence in measure. Specifically, we generalize the theorem for the casewhere the codomain is a separable metric space and for the case where the limiting map isconstant and the codomain is an arbitrary topological space.
Keywords: almost everywhere convergence; asymptotic statistical inference; convergencein measure; separability
MSC 2010:
It is a classical result that, if f, f , f , . . . are measurable C -valued functions on a finite measurespace and if f n → f almost everywhere, then f n → f in measure. The importance of theconvergence theorem is fully aware. It would be useful (and also intellectually amusing) to provethe convergence theorem when the codomain of the maps f, f , f , . . . is a metric space or evenan arbitrary topological space. This task is not trivial; for example, the usual proof approach,for f constant, cannot deal with the case when C is replaced with an arbitrary topological space.We give a new proof for the convergence theorem that, to a certain extent, allows of theaforementioned generalization. At the same time, although an application of our generalization,for purposes such as a probabilistic or statistical one, is in a sense immediate for “well-behaved”maps as a probability measure is a suitably scaled finite measure, we provide a counterexampleshowing that the result does not necessarily hold if the measurability of the involved mapsis undecided; difficulty in proving measurability is not unusual in applications such as in thecontext of asymptotic statistical inference, e.g. establishing the measurability of a nonlinearleast squares estimator in R k for some integer k ≥ Throughout, we fix a finite measure space (Ω , F , M ).Following the convention of probability theory, we will in general write for simplicity a set ofthe form { ω | g ( ω ) has a given property } as { g has the property } . When written in juxtaposition ∗ Author for correspondence: Yu-Lin Chou, Institute of Statistics, National Tsing Hua University, Hsinchu30013, Taiwan; Email: [email protected] . The author would like to gratefully acknowledge the con-structive comments obtained for the previous drafts of the present paper. { g has the property } will simply take the form ( g has the property).If S is a topological space with B S the Borel sigma-algebra, if f n : Ω → S is ( F , B S )-measurable for all n ∈ N , and if f : Ω → S is constant, then, regarding the convergence of thesequence ( f n ) to f , the involved notion of closeness to f is understood in terms of the opensubsets of S that contain (the point of S identified with) f . For example, the definition ofconvergence in M -measure is to be paraphrased in this case as “for every open G ⊂ S containingthe constant identified with f , we have M ( { ω ∈ Ω | f n ( ω ) / ∈ G } ) = M ( f n / ∈ G ) → Given a sequence of subsets of Ω, we can partition the space Ω into the limit inferior of thesequence and the limit superior of the sequence of the complements of the subsets of Ω; thisobservation is the fundamental proof idea.
Theorem 1.
Let S be a topological space; let f, f n : Ω → S be measurable- ( F , B S ) for all n ∈ N ; let f n → f almost everywhere with respect to M . i) If S is in particular a separablemetric space, then f n → f in M -measure; ii) if f is in particular a constant map, then f n → f in M -measure.Proof. Let ε > d be the separable metric on S × S . Since d is continuous with respect to theproduct d -topology, and since the countable base property of S ensures that the map ( f n , f ) ismeasurable with respect to the Borel sigma-algebra generated by the product d -topology for all n ∈ N , the function d ( f n , f ) is measurable for all n ∈ N .Let N ∈ N whenever lim inf n →∞ { d ( f n , f ) ≤ ε } is empty; otherwise, let N := inf ω ∈ Ω inf J ,where, for every ω ∈ Ω, the inner infimum extends over all J ∈ N such that d ( f j ( ω ) , f ( ω )) ≤ ε for all j ≥ J . Then, for all n ≥ N we have0 ≤ M ( d ( f n , f ) > ε )= M (cid:0) d ( f n , f ) > ε, d ( f m , f ) ≤ ε for sufficiently large m (cid:1) + M (cid:0) d ( f n , f ) > ε, lim sup m →∞ d ( f m , f ) > ε (cid:1) ≤ M (cid:0) lim sup m →∞ d ( f m , f ) > ε (cid:1) = M (Ω) − M (cid:0) d ( f m , f ) ≤ ε for sufficiently large m (cid:1) ≤ M (Ω) − M ( f n → f pointwise)= 0;the last equality follows from the convergence assumption.For ii), we rewrite the partition of the finite measure space Ω used above as { lim inf n →∞ { f n ∈ G } , lim sup n →∞ { f n / ∈ G }} with G ∋ f being a given open subset of S . Then ii) follows from the main argument above withthe apparent slight modification. Remark.
For the breadth of some application possibilities of Theorem 1, we recall that manyfamiliar function spaces can be made a separable metric space, e.g. the space R ∞ (equipped with2 usual product metric), the spaces L p ( R n ) (equipped with the metric induced by the usual L p -norm) for 1 ≤ p < + ∞ and n ∈ N , the space C of all the R -valued continuous functions on [0 , D of all the R -valued c´adl´ag functions on[0 ,
1] (equipped with the Skorokhod metric, which is a metric derived from the uniform metric),are separable metric spaces. If the involved maps f, f n are not all measurable, or if the measurability is not obvious, thenone may try to circumvent the measurability issue via the outer measure obtained by taking forevery A ⊂ Ω the infimum of the set { M ( B ) | B ⊃ A, B ∈ F } and consider the convergencemodes in terms of the M -outer measure. The convergence modes with respect to the M -outermeasure reduce to the usual modes, respectively, whenever measurability is available.However, even with the M -outer measure, the first conclusion of Theorem 1 does not ne-cessarily hold in the presence of a measurability issue. Indeed, a consideration over rationaltranslations of a usual Vitali set V (which is not Lebesgue measurable) in [0 , × A ∈ [0 , /R A where the productextends over all the elements of the quotient space [0 , /R with respect to the equivalence re-lation R ⊂ [0 , defined by declaring that xRy iff x − y ∈ Q , with full outer measure wouldlead to a counterexample. Here we certainly acknowledge the axiom of choice. Although thecounterexample thus obtained is somewhat of a routine nature, for clarity we still elaborate andhighlight a possible construction: Proposition 1.
There are some Borel finite measure space and some sequence of nonmeasurablefunctions from the measure space to R that converges pointwise but not in the outer measure.Proof. Consider the unit interval [0 ,
1] equipped with Lebesgue measure L restricted to the Borelsubsets of [0 , V ⊂ [0 ,
1] such that L ∗ ( V ) = 1. Let { q n } n ∈ N := Q ∩ ]0 , n ∈ N , let V n be obtained from a rational translation V + q n of V so that { V n } is a partition of [0 , V n is not Lebesgue measurable, and each V n has Lebesgue-outermeasure L ∗ ( V n ) = 1.Let f n := V n on [0 ,
1] for each n ∈ N ; then each f n is not Lebesgue measurable, and f n → f n → L and L ∗ .However, we have L ∗ ( | f n | > ε ) = L ∗ ( V n ) = 1 for all n ∈ N and all 0 < ε <
1; so the sequence( f n ) does not converge in L ∗ to the zero function.Since the measure space [0 ,
1] considered in the above proof can be viewed as the probabilityspace describing the uniform distribution concentrated on [0 , By a c´adl´ag function we mean a function that has left limit and is right continuous everywhere. For the separability of L p ( R n ), there is a proof given in Brezis [2]; for the separability of each of the othercases, there is a proof contained in Billingsley [1]. eferences [1] Billingsley, P. (1999). Convergence of Probability Measures , second edition. John Wiley &Sons, Chichester.[2] Brezis, H. (2011).
Functional Analysis, Sobolev Spaces and Partial Differential Equations .Springer, New York.[3] Jennrich, R. I. (1969). Asymptotic properties of nonlinear least squares estimators.
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