(n) 4) Let E be measurable and f: E→ R be measurable. Let y=k/n for ke Z and ne N. Let Ik =)) so that clearly R = (). Let E() = f((n)) and define (n) 00 (n) On(x) = x()(x). k=-∞ Prove that each on is measurable and that on →f uniformly on R. Hint: = For the former, if on,N(x) Σk-N-1X(n) (x) then prove that we have lim No On,Non pointwise on E. Why is each on,N measurable? 4

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Question
4
(n)
4) Let E be measurable and f: E→ R be measurable. Let y=k/n for ke Z and
ne N. Let Ik =)) so that clearly R = (). Let E() = f((n)) and define
(n)
00
(n)
On(x) = x()(x).
k=-∞
Prove that each on is measurable and that on →f uniformly on R.
Hint:
=
For the former, if on,N(x) Σk-N-1X(n) (x) then prove that we have
lim No On,Non pointwise on E. Why is each on,N measurable?
4
Transcribed Image Text:(n) 4) Let E be measurable and f: E→ R be measurable. Let y=k/n for ke Z and ne N. Let Ik =)) so that clearly R = (). Let E() = f((n)) and define (n) 00 (n) On(x) = x()(x). k=-∞ Prove that each on is measurable and that on →f uniformly on R. Hint: = For the former, if on,N(x) Σk-N-1X(n) (x) then prove that we have lim No On,Non pointwise on E. Why is each on,N measurable? 4
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