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2.2.2 Monotone Convergence Theorem

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2.2.2 Monotone Convergence Theorem CHAPTER 2. THE LEBESGUE INTEGRAL I 22 2.2.2 Monotone convergence theorem The following theorem is what is known in the literature as the "monotone con- vergence theorem" which is a statement about nonnegative increasing sequences (fn)n≥1 of measurable functions in a measure space (Ω; Σ; µ). It has a Corollary which we call here the monotone convergence theorem for summable/integrable 1 functions where the assumption fn ≥ 0 a.e. is replaced by fn 2 L . Contrary to the monotone convergence theorem for the Riemann integral the measurability or integrability of the pointwise (or a.e.) limit is part of the conclusion and not of an hypothesis to be put into the theorem. Monotone convergence theorem - general version: measurable func- + tions. If (fn)n≥1 is a monotone increasing sequence of functions in M (Ω; Σ) which converges almost everywhere to a function f on Ω. Then f is measurable and Z Z Z lim fn(x) dµ = f(x) dµ = lim fn(x) dµ 2 [0; +1]: (2.24) n!1 Ω Ω Ω n!1 R In particular, if lim fn(x) µ(dx) < +1, then f is summable/integrable. n!1 Ω Remark. Note the monotonicity allows us to define f(x) := lim fn(x) 2 [0; +1] n!1 Remark. When the fn are summable/integrable we can drop the assumption that the fn ≥ 0 by considering in that case the non-negative sequence gn = fn − f1. This fact is a corollary of the monotone convergence theorem but we state it as monotone convergence for integrable functions Corollary: Monotone convergence - integrable functions. If (fn)n≥1 is a monotone increasing sequence of functions in L1(Ω; dµ) which converges almost everywhere to a function f on Ω. Then f is measurable and Z Z Z lim fn(x) dµ = f(x) dµ = lim fn(x) dµ 2 [0; +1]: (2.25) n!1 Ω Ω Ω n!1 R In particular, if lim fn(x) µ(dx) < +1, then f is summable/integrable. n!1 Ω Proof of the monotone convergence theorem. First of all, we prove the theorem in the case that fn converges pointwise to f. By the monotonicity of the sequence Sfn (t) ⊂ Sfn+1 (t) for all t and n. By definition 1 [ Sf (t) = Sfn (t) n=1 CHAPTER 2. THE LEBESGUE INTEGRAL I 23 and therefore by the continuity of measures 1 [ lim µ(Sf (t)) = µ Sf (t) = µ(Sf (t)): n!1 n n n=1 note that this convergence is monotone for a sequence of monotone functions. Now the theorem follows from the corresponding theorem of Riemann integra- tion. Indeed, let νn(t) = µ(Sfn (t)) and ν(t) = µ(Sf (t)). Then νn(t) ≤ νn+1(t) ≤ ν(t) for all n 2 N and t > 0 and all functions are decreasing functions of t, hence Riemann integrable on any compact interval. Since obviously Z 1 Z 1 lim νn(t) dt ≤ ν(t) dt n!1 0 0 we have to prove the reversed inequality in the case when the sequence In = R 1 0 νn(t) dt is bounded which implies that νn(t) < 1 and therefore ν(t) < 1 for all t > 0 . Let b > a > 0. Since the νn(t) are decreasing functions of t we have for any positive integer N that N Z b Z b b − a X k(b − a) ν (t) dt ≥ ν (t) dt ≥ ν (a ); a = a + n n N n k k N 0 a k=1 Since ν(t) is a decreasing function of t for any > 0 there is a positive integer N such that N Z b b − a X ν(t) dt ≤ ν(a ) + . N k a k=1 Since νn(t) converges pointwise to ν(t) for all n sufficiently large max jνn(ak) − ν(ak)j < /(b − a) k2f1;:::;Ng with N as selected before. This implies that for all > 0 and all n sufficiently large Z b Z b νn(t) dt ≥ ν(t) dt − 2. a a which implies Z b Z b lim νn(t) dt ≥ ν(t) dt n!1 a a and this inequality proves the claim. Finally we prove the monotone convergence theorem when fn converges to f almost everywhere. We need the following lemma. Lemma 1. Let f 2 M +(Ω; Σ). Then f(x) = 0 almost everywhere if and only if R f dµ = 0. Proof. If R f dµ = 0, then for all a > 0. Z 1 Z a 0 = µ(Sf (t)) dt ≥ µ(Sf (t)) dt ≥ aµ(Sf (a)) 0 0 CHAPTER 2. THE LEBESGUE INTEGRAL I 24 where the latter inequality is called Chebychev's inequality. Hence µ(Sf (a)) = 0 S 1 for all a > 0 Put a = 1=n and note that Sf (0) = Sf ( n ) which has measure n zero thanks to the sigma-additivity of µ. Conversely, let f(x) = 0 almost everywhere. Then µ(Sf (t)) = 0 for all t ≥ 0 and therefore R f dµ = 0 proving the lemma. Proof of the monotone convergence theorem in the case of almost everywhere convergence. Convergence almost everywhere means that there exists a set N ⊂ Ω of measure zero such that fn converges pointwise to f on E := ΩnN. We then apply the monotone convergence theorem to fχE. Indeed, with Lemma 1 we have Z Z Z Z f dµ = fχE dµ = lim fnχE dµ = lim fn dµ. n!1 n!1 2.2.3 Fatou's lemma Fatou's lemma is not only needed to prove the dominated convergence theorem below but it includes also a statement of the behaviour of the integral under pointwise (or a.e.) convergence: The integral is lower semi-continuous under a.e. convergence. + Fatou's lemma. If (fn)n is a sequence of functions in M (Ω; Σ). Then f(x) := lim inffn(x) is measurable and n!1 Z Z Z lim inf fn(x) µ(dx) ≥ f(x) µ(dx) = lim inffn(x) µ(dx) : (2.26) n!1 Ω Ω Ω n!1 In particular, if (fn)n is a sequence of summable functions, then f is summable. Remark. The hypothesis that the fn's are nonnegative is crucial (see exer- cises). Remark. Fatou's lemma means that the mapping f 7! R f is pointwise lower semicontinuous. Corollary. If f(x) = lim fn(x) a.e. , then n!1 Z Z lim inf fn(x) µ(dx) ≥ f(x) µ(dx): n!1 Ω Ω Proof of Fatou's lemma. We have already shown before that f is measur- able. For k ≥ 1 let Fk(x) = infn≥k fn(x). Fk is measurable and if all fn are summable it is also summable since Fk ≤ fk and more generally Fk ≤ fn if n ≥ k. Therefore for all n ≥ k Z Z fn(x) µ(dx) ≥ Fk(x) µ(dx): Ω Ω CHAPTER 2. THE LEBESGUE INTEGRAL I 25 Obviously, Fk is an increasing sequence (i.e Fk+1 ≥ Fk with lim Fk(x) = sup( inf fn(x)) =: lim inffn(x) = supFk(x): k!1 k≥1 n≥k n!1 k≥1 Consequently, applying the monotonicity of the integral and the monotone con- vergence theorem, we have Z Z lim inf fn(x) µ(dx) := sup inf fn(x) µ(dx) n!1 Ω k≥1 n≥k Ω Z Z ≥ lim Fk(x) µ(dx) = f(x) µ(dx): k!1 Ω Ω 2.2.4 Dominated convergence theorem The dominated convergence theorem formulates sufficient conditions under which almost everywhere convergence yields an integrable function and limit and inte- gral are interchangeable. Again this is an important difference between Lebesgue and Riemann integral. Dominated convergence theorem. Let (fn)n be a sequence of (summable) functions in M(Ω; Σ) converging a.e. to a function f. If there exists a nonneg- ative summable function g on (Ω; Σ; µ) such that jfn(x)j ≤ g(x) for all n, then f is summable such that jf(x)j ≤ g(x) and Z Z Z lim fn(x) µ(dx) = f(x) µ(dx) = lim fn(x) µ(dx) : (2.27) n!1 Ω Ω Ω n!1 Proof. Since g − fn ≥ 0 and g + fn ≥ 0 we may apply Fatou's lemma to both sequences of functions: Z Z Z Z g(x)−f(x) µ(dx) ≤ lim inf g(x)−fn(x) µ(dx) ≤ g(x) µ(dx)−lim sup fn(x) µ(dx); Ω n!1 Ω Ω n!1 Ω Z Z Z Z g(x)+f(x) µ(dx) ≤ lim inf g(x)+fn(x) µ(dx) ≤ g(x) µ(dx)+lim inf fn(x) µ(dx): Ω n!1 Ω Ω n!1 Ω Consequently, Z Z Z lim sup fn(x) µ(dx) ≤ f(x) µ(dx) ≤ lim inf fn(x) µ(dx); n!1 Ω Ω n!1 Ω which implies that Z Z lim fn(x) µ(dx) = f(x) µ(dx): n!1 Ω Ω Corollary - convergence in L1. Under the assumptions of the dominated 1 convergence theorem the sequence (fn)n converges in L (Ω; dµ), that is Z lim jfn(x) − f(x)j µ(dx) = 0: (2.28) n!1 Ω CHAPTER 2. THE LEBESGUE INTEGRAL I 26 Proof. Since the sequence (jfn −fj)n also satisfies the hypotheses of the dom- inated convergence theorem (in particular, the jfn − fj are summable since f is summable by the dominated convergence theorem). Now jfn − fj ! 0 a.e.
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