This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View Full
Document
 Fall '09
 HEIL

Click to edit the document details