section2.4b - REAL ANALYSIS LECTURE NOTES 2.4 MODES OF...

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REAL ANALYSIS LECTURE NOTES: 2.4 MODES OF CONVERGENCE CHRISTOPHER HEIL 2.4.1 The relation between convergence in measure and pointwise convergence Although convergence in measure does not imply pointwise convergence, we do have the following weaker (but still very useful) conclusion. Theorem 1. If f n m f , then there exists a subsequence { f n k } k N such that f n k f point- wise a.e. Proof. Since f n m f , we can find n 1 < n 2 < · · · such that n n k , μ n | f - f n | > 1 k o 1 2 k . Define E k = n | f - f n k | > 1 k o and H m = S k = m E k . Then we have μ ( E k ) < 1 2 k and μ ( H m ) X k = m 1 2 k = 1 2 m - 1 . Set Z = T m =1 H m . Then μ ( Z ) μ ( H m ) 1 / 2 m - 1 for every m , so we have μ ( Z ) = 0. If x / Z , then x / H m for some m . Hence x / E k for all k m , which implies | f ( x ) - f n k ( x ) | ≤ 1 k , all k m. Thus f n k ( x ) f ( x ) for all x / Z . Since Z has measure zero, we therefore have pointwise convergence of f n k to f almost everywhere. As an important special case we have the following. Corollary 2. If f n f in L 1 ( X ), then there exists a subsequence { f n k } k N such that f n k f pointwise a.e. These notes follow and expand on the text “Real Analysis: Modern Techniques and their Applications,” 2nd ed., by G. Folland. 1
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2 2.4 MODES OF CONVERGENCE 2.4.2 A Cauchy criterion for convergence in measure Although convergence in measure is not associated with a particular norm, there is still a useful Cauchy criterion for convergence in measure. Definition 3. Given measurable f n on X , we say that { f n } n Z is Cauchy in measure if ε > 0 , μ {| f m - f n | ≥ ε } → 0 as m, n → ∞ . Precisely, this means that ε, η > 0 , N > 0 such that m, n > N = μ {| f m - f n | ≥ ε } < η. The usefulness of the Cauchy criterion is that it does not require us to know what the limit function is — we can test for Cauchyness without knowing whether the sequence converges. Further, the next theorem says that for convergence in measure, Cauchyness is equivalent to convergence. In order to prove the theorem, we need the following exercise. Exercise 4. Let f n be measurable functions on X . (a) Prove that E = { x X : lim n →∞ f n ( x ) exists } is measurable. (b) Show that f ( x ) = lim n →∞ f n ( x ) , if the limit exists , 0 , otherwise , is a measurable function. Theorem 5. Given measurable f n on X , the following statements are equivalent. (a) { f n } n N is Cauchy in measure. (b) There exists an f such that f n m f .
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