5) Let E be measurable with m(E) < ∞. Suppose each fr : ER is measurable and fnf pointwise on E to some f: E R. Given any e > 0 and 8 > 0, prove (with- out using Egoroff's theorem) that there exists ACE and NE N where m(E\A) < € and \fn(x) f(x)\<8 for every Є A and n > N. Hint: Consider the sets AN = {x = E:\fn(x) - f(x)| < 6 for all n > N}. Prove these are ascending, UN-1AN E, and use continuity of measure. Now simply set A = AN for large enough N! =
5) Let E be measurable with m(E) < ∞. Suppose each fr : ER is measurable and fnf pointwise on E to some f: E R. Given any e > 0 and 8 > 0, prove (with- out using Egoroff's theorem) that there exists ACE and NE N where m(E\A) < € and \fn(x) f(x)\<8 for every Є A and n > N. Hint: Consider the sets AN = {x = E:\fn(x) - f(x)| < 6 for all n > N}. Prove these are ascending, UN-1AN E, and use continuity of measure. Now simply set A = AN for large enough N! =
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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Transcribed Image Text:5) Let E be measurable with m(E) < ∞. Suppose each fr : ER is measurable and
fnf pointwise on E to some f: E R. Given any e > 0 and 8 > 0, prove (with-
out using Egoroff's theorem) that there exists ACE and NE N where m(E\A) < € and
\fn(x) f(x)\<8 for every Є A and n > N.
Hint: Consider the sets AN = {x = E:\fn(x) - f(x)| < 6 for all n > N}. Prove these
are ascending, UN-1AN E, and use continuity of measure. Now simply set A = AN for
large enough N!
=
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