b. there is a natural number for all x € Ce. Let f [0, 1] (0, 1] be a measurable function. Show that for every € > 0, : N, and a set Ce such that m(Ce) < € and 1 Ne +1 < f(x) ≤ № c
b. there is a natural number for all x € Ce. Let f [0, 1] (0, 1] be a measurable function. Show that for every € > 0, : N, and a set Ce such that m(Ce) < € and 1 Ne +1 < f(x) ≤ № c
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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Question
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kindly answer letter b.
![2. Let ([0, 1], L, m) be a Lebesgue measure space, and let A be a nonempty measurable
subset of [0, 1]. Let {E}
[0, 1] be a countable disjoint collection of Lebesgue
measurable sets.
a.
Show that
m (AnŮ Ex) = m(
k=1
k=1
Σm(An Ek).
b.
Let f [0, 1] (0, 1] be a measurable function. Show that for every € > 0,
there is a natural number N and a set C such that m(Ce) < € and
< f(x) ≤ / / /
for all x € Ce.
1
1
NE+1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2f3f89d-00cb-4f18-b8de-1c4d3b4cc1ef%2F1a544b40-e5b3-48d6-96b1-21257d42b51a%2Fg1xhe6n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let ([0, 1], L, m) be a Lebesgue measure space, and let A be a nonempty measurable
subset of [0, 1]. Let {E}
[0, 1] be a countable disjoint collection of Lebesgue
measurable sets.
a.
Show that
m (AnŮ Ex) = m(
k=1
k=1
Σm(An Ek).
b.
Let f [0, 1] (0, 1] be a measurable function. Show that for every € > 0,
there is a natural number N and a set C such that m(Ce) < € and
< f(x) ≤ / / /
for all x € Ce.
1
1
NE+1
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