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lusin - FUNCTIONAL ANALYSIS LECTURE NOTES EGOROFF AND...

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FUNCTIONAL ANALYSIS LECTURE NOTES: EGOROFF AND LUSIN’S THEOREMS CHRISTOPHER HEIL 1. Egoroff’s Theorem Egoroff’s Theorem is a useful fact that applies to general bounded positive measures. Theorem 1 (Egoroff’s Theorem) . Suppose that μ is a finite measure on a measure space X, and f n , f : X C are measurable. If f n f pointwise a.e., then for every ε > 0 there exists a measurable E X such that (a) μ ( E ) < ε , and (b) f n converges uniformly to f on E C = X \ E , i.e., lim n →∞ sup x/ E | f ( x ) f n ( x ) | = 0 . Proof. Let Z be the set of measure zero where f n ( x ) does not converge to f ( x ). For k , n N , define the measurable sets E n ( k ) = uniontext m = n braceleftBig | f f m | ≥ 1 k bracerightBig and Z k = intersectiontext n =1 E n ( k ) . Now, if x Z k , then x E n ( k ) for every n . Hence, for each n there must exist an m n such that | f ( x ) f m ( x ) | > 1 k . Therefore f n ( x ) does not converge to f ( x ), so x Z . Thus Z k Z, and therefore μ ( Z k ) = 0 by monotonicity. Since E 1 ( k ) E 2 ( k ) ⊇ · · · , we therefore have by continuity from above that lim n →∞ μ ( E n ( k )) = μ ( Z k ) = 0 . Choose now any ε > 0. Then for each k , we can find an n k such that μ ( E n k ( k )) < ε 2 k . Define E = uniontext k =1 E n k ( k ) , c circlecopyrt 2009 by Christopher Heil. 1

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2 EGOROFF’S AND LUSIN’S THEOREMS then we have by subadditivity that μ ( E ) ε . And if x / E , then x / E n k ( k ) for every k , and therefore | f ( x ) f m ( x ) | < 1 k for all m n k . Thus, we have shown that for each
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lusin - FUNCTIONAL ANALYSIS LECTURE NOTES EGOROFF AND...

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