Tel Aviv University, 2015 Functions of real variables 46 5 Modes of convergence 5a Introduction . 46 5b Local convergence in measure . 47 5c Convergence almost everywhere . 49 5d Dominated convergence; integrals depending on a parameter . 51 5e One-sided bounds . 53 5f Spaces Lp ....................... 56 . 5a Introduction If one has a sequence x1; x2; x3; · · · 2 R of real numbers xn, it is unambiguous what it means for that sequence to converge to a limit x 2 R . If, however, one has a sequence f1; f2; f3;::: of functions fn : X ! R . and a putative limit f : X ! R . there can now be many different ways in which the sequence fn may or may not converge to the limit f . Once X becomes infinite, the functions fn acquire an infinite number of degrees of freedom, and this allows them to approach f in any number of inequivalent ways. However, pointwise and uniform conver- gence are only two of dozens of many other modes of convergence that are of importance in analysis. Tao, Sect. 1.5 \Modes of convergence". When speaking about functions fn on a measure space we really mean, as usual, equivalence classes [fn]. Thus we restrict ourselves to such modes of convergence that are insensitive to arbitrary change of fn on a null set. (Do you bother whether the null set depends on n?) We replace the pointwise convergence with the convergence almost everywhere, fn ! f a.e. () µfx : fn(x) 6! f(x)g = 0 () () fn ! f pointwise outside some null set; and the uniform convergence with the convergence uniform almost every- where, fn ! f in L1 () fn ! f uniformly outside some null set Tel Aviv University, 2015 Functions of real variables 47 (why \L1"? wait for 5f7). On the other hand, the given measure leads to new modes of convergence, such as Z fn ! f in L1 () jfn − fj dµ ! 0 : X 5b Local convergence in measure This is the weakest among all modes of convergence that we need. (That is, every other convergence implies the local convergence in measure.) Strangely enough, it is not mentioned by Tao (Sect. 1.5, p. 95) among 5 modes.1 Throughout, (X; S; µ) is a σ-finite measure space. 5b1 Definition. Let f; f1; f2; ··· : X ! R be measurable functions. We say that fn ! f locally in measure, if µfx 2 A : jfn(x) − f(x)j ≥ "g ! 0 as n ! 1 for all " > 0 and all A 2 S such that µ(A) < 1. This definition generalizes readily to fn : X ! [−∞; +1]; but f must be finite a.e. Clearly, fn ! f if and only if fn − f ! 0. 5b2 Exercise. It is enough to check a single sequence of sets A = A1;A2;::: such that [kAk = X. Prove it.2 If µ(X) < 1, then it is enough to check A = X, of course; this case is called the global convergence in measure. For µ(X) = 1 the global version is stronger than the local one; for example, on R, 1l[n;1) ! 0 locally but not globally. 5b3 Remark. If fn ! f and fn ! g then f = g a.e. (Local convergence in measure is meant in both cases.) Proof (sketch): fx : jf(x) − g(x)j ≥ 2"g ⊂ fx : jfn(x) − f(x)j ≥ "g [ fx : jfn(x) − g(x)j ≥ "g. 1Also not mentioned by Capi´nski& Kopp, nor by Jones. Mentioned by N. Lerner \A course on integration theory", Springer 2014 (Exercise 2.8.14 on p. 113); by F. Liese & K.-J. Miescke \Statistical decision theory", Springer 2008 (Def. A.11 on p. 619); and, prominently, on Wikipedia, \Convergence in measure". 2 Hint: µ A n (A1 [···[ An) ! 0. Tel Aviv University, 2015 Functions of real variables 48 5b4 Remark (sandwich). If gn ≤ fn ≤ hn a.e., gn ! f and hn ! f, then fn ! f. (Local convergence in measure is meant in all three cases.) Proof (sketch): fx : jfn(x) − f(x)j ≥ "g ⊂ fx : jgn(x) − f(x)j ≥ "g [ fx : jhn(x) − f(x)j ≥ "g. 5b5 Exercise. If fn ! 0 and gn ! 0, then afn + bgn ! 0 for arbitrary a; b 2 R. (Local convergence in measure is meant in all three cases; no matter what is meant by 1 − 1.) Prove it. Thus, if fn ! f and gn ! g, then afn + bgn ! af + bg (since (afn + bgn) − (af + bg) = a(fn − f) + b(gn − g)). Interestingly, the local convergence in measure is insensitive not only to the choice of functions fn in their equivalence classes, but also to the choice of a measure µ in its equivalence class defined as follows. 5b6 Definition. A measure ν on (X; S) is called equivalent to µ if ν = f · µ for some f : X ! (0; 1). This is indeed an equivalence relation; symmetry: if ν = f · µ then 1 µ = f · ν; transitivity: if ν1 = f · µ and ν2 = g · ν1 then ν2 = (fg) · µ (recall 4c26 and the paragraph after it). If µ and ν are equivalent then they have the same null sets (think, why). 5b7 Lemma. If ν is equivalent to µ, then fn ! f locally in µ if and only if fn ! f locally in ν. Proof. Let fn ! f locally in µ, " > 0 and ν(A) < 1; we have to prove that ν(An) ! 0, where An = fx 2 A : jfn(x) − f(x)j ≥ "g. By 5b2, WLOG we dν dµ may assume that dµ and dν are bounded on A, since countably many such dµ sets A can cover X (think, why). Now, µ(A) ≤ supx2A dν (x) ν(A) < 1, dν thus µ(An) ! 0, and ν(An) ≤ supx2A dµ (x) µ(An) ! 0. 5b8 Exercise. There exists a finite measure ν equivalent to µ. Prove it.1 For such ν, local convergence in µ is equivalent to the global convergence in ν. The latter is easy to metrize. 5b9 Exercise. If µ(X) < 1, then the following conditions on measurable fn : X ! R are equivalent: (a) fn ! 0 in measure; 1 dν Hint: X = ]Ak, dµ = "k on Ak. Tel Aviv University, 2015 Functions of real variables 49 R (b) X min(1; jfnj) dµ ! 0; (c) R jfnj dµ ! 0. X 1+jfnj Prove it. 1 We may define a metric ρ on the set L0(X) of all equivalence classes of measurable functions X ! R by2 Z (5b10) ρ[f]; [g] = min(1; jf − gj) dµ X provided that µ(X) < 1; otherwise we use a finite equivalent measure in- stead. 5c Convergence almost everywhere As was noted in Sect. 4c, \almost everywhere" (a.e.) means \outside some null set". In particular, we say that fn ! f a.e., if fn(x) ! f(x) for all x outside some null set. (Sometimes this convergence is called \pointwise a.e.".) Once again, the definition generalizes readily to fn : X ! [−∞; +1]; but f must be finite a.e. Once again, fn ! f if and only if fn − f ! 0. Also, if fn ! f and gn ! g, then afn + bgn ! af + bg (no matter what is meant by 1 − 1). Still, (X; S; µ) is a σ-finite measure space. 5c1 Lemma. Convergence almost everywhere implies local convergence in measure. Proof. First, we consider monotone convergence: fn # 0 a.e. In this case, given " > 0 and µ(A) < 1, the sets An = fx 2 A : fn(x) ≥ "g satisfy An #;, that is, A n An " A (up to a null set), which implies µ(A n An) " µ(A), that is, µ(An) # 0. Second, the general case: fn ! 0 a.e. We introduce functions gn = inf(fn; fn+1;::: ), hn = sup(fn; fn+1;::: ); they are measurable by 3c10. Al- 3 most everywhere, gn " 0 (think, why), hn # 0, and gn ≤ fn ≤ hn. By the first part of this proof, gn ! 0 and hn ! 0 locally in measure. It remains to use 5b4. 5c2 Corollary (of 5c1 and 5b3). If fn ! f locally in measure and fn ! g a.e., then f = g a.e. 1 0 Denoted also by L0(X; S; µ), L0(µ), L (X), etc. 2 R jf−gj Alternatively, use X 1+jf−gj dµ. 3Note the zigzag sandwich! Tel Aviv University, 2015 Functions of real variables 50 5c3 Example. Local convergence in measure does not imply convergence almost everywhere. We take the measure space [0; 1] with Lebesgue measure. For every n there exists a partition of X = [0; 1] into n sets (just intervals, if you like) of measure 1=n each. We combine such partitions (for odd n) into a single infinite sequence of sets A1;A2;A3; A4;A5;A6;A7;A8; A9;A10;A11;A12;A13;A14;A15; ::: | {z } | {z } | {z } partition partition partition such that Ak2 ] Ak2+1 ]···] A(k+1)2−1 = X and µ(Ak2 ) = µ(Ak2+1) = 1 ··· = µ(A(k+1)2−1) = 2k+1 for k = 1; 2;::: Clearly, µ(An) ! 0, therefore the indicators fn = 1lAn converge to 0 in measure. However, lim supn fn(x) = 1 for every x, since x belongs to infinitely many An. 5c4 Exercise. If X is (finite or) countable, then local convergence in measure implies convergence almost everywhere. Prove it. P 5c5 Lemma. Let µ(X) < 1, and ρ be defined by (5b10). If n ρ(fn; fn+1) < 1, then the sequence (fn)n converges a.e. Proof. We denote gn = min(1; jfn+1 − fnj), Sn = g1 + ··· + gn " S : X ! [0; 1]. Using the Monotone Convergence Theorem 4c11, Z Z Z Z X Z S dµ = lim Sn dµ = lim g1 dµ+···+ gn dµ = gn dµ < 1 : n n X X X X n X P By (4c31), S < 1 a.e.