Please see attached image...Egoroff's Theorem - Assume E has finite measure. Let {fn} be a sequence of measurable functions on E that converges pointwise on E to the rea-valued function f. Then for each ε > 0, there is a closed set F contained in E for which {fn} → f uniformly on F and m(E ~ F) < ε. Suppose that E is a measurable set of finite measure and that (fn) is a sequenceof measurable functions on E that converges pointwise to a fon E. Show that there existmeasurable sets {Ek : k = 0, 1, 2, ...}, where m(Eo) = 0, andEkE and, for eachk=0k = 1, 2, 3, the sequence (fn) converges uniformly to fon E. (Hint: Use Egoroff'sTheorem)=

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Suppose that E is a measurable set of finite measure and that (fn) is a sequence of measurable functions on E that converges pointwise to a fon E. Show that there exist measurable sets {E: k = 0, 1, 2,...}, where m(Eo) = 0, and Ek = E and, for each k=0 k = 1, 2, 3, ... the sequence (fn) converges uniformly to fon Ek. (Hint: Use Egoroff's Theorem)

Suppose that E is a measurable set of finite measure and that (fn) is a sequence of measurable functions on E that converges pointwise to a fon E. Show that there exist measurable sets {E: k = 0, 1, 2,...}, where m(Eo) = 0, and Ek = E and, for each k=0 k = 1, 2, 3, ... the sequence (fn) converges uniformly to fon Ek. (Hint: Use Egoroff's Theorem)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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