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Unformatted text preview: FUNCTIONAL ANALYSIS LECTURE NOTES: EGOROFF AND LUSINS THEOREMS CHRISTOPHER HEIL 1. Egoroffs Theorem Egoroffs Theorem is a useful fact that applies to general bounded positive measures. Theorem 1 (Egoroffs Theorem) . Suppose that 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 > 0 there exists a measurable E X such that (a) ( E ) < , 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 ( Z k ) = 0 by monotonicity. Since E 1 ( k ) E 2 ( k ) , we therefore have by continuity from above that lim n ( E n ( k )) = ( Z k ) = 0 . Choose now any > 0. Then for each k , we can find an n k such that ( E n k ( k )) < 2 k ....
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- Fall '09