2. Let ([0, 1], L, m) be a Lebesgue measure space, and let A be a nonempty measurable subset of [0, 1]. Let {E}1 [0, 1] be a countable disjoint collection of Lebesgue measurable sets. Show that a. m(AnŪR) - Σm(ANB). Ex) Σm(ΑΠΕ). k=1 k=1 Let f [0, 1] (0, 1] be a measurable function. Show that for every € > 0, N and a set C such that m(C) < € and < f(x) ≤ NE+1 1 Ne b. there is a natural number for all x € Ce.
2. Let ([0, 1], L, m) be a Lebesgue measure space, and let A be a nonempty measurable subset of [0, 1]. Let {E}1 [0, 1] be a countable disjoint collection of Lebesgue measurable sets. Show that a. m(AnŪR) - Σm(ANB). Ex) Σm(ΑΠΕ). k=1 k=1 Let f [0, 1] (0, 1] be a measurable function. Show that for every € > 0, N and a set C such that m(C) < € and < f(x) ≤ NE+1 1 Ne b. there is a natural number for all x € Ce.
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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ANSWER B ONLY!!
![2. Let ([0, 1], L, m) be a Lebesgue measure space, and let A be a nonempty measurable
subset of [0, 1]. Let {E}1 [0, 1] be a countable disjoint collection of Lebesgue
measurable sets.
Show that
a.
m(AnŪR) - Σm(ANB).
Ex)
Σm(ΑΠΕ).
k=1
k=1
Let f [0, 1] (0, 1] be a measurable function. Show that for every € > 0,
N and a set C such that m(C) < € and < f(x) ≤
NE+1
1
Ne
b.
there is a natural number
for all x € Ce.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4228d8f-8dd1-458e-9871-4eecf6e388c5%2F24eade1c-54e7-4587-883f-6807dd7ad281%2F89gmsvg_processed.png&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}1 [0, 1] be a countable disjoint collection of Lebesgue
measurable sets.
Show that
a.
m(AnŪR) - Σm(ANB).
Ex)
Σm(ΑΠΕ).
k=1
k=1
Let f [0, 1] (0, 1] be a measurable function. Show that for every € > 0,
N and a set C such that m(C) < € and < f(x) ≤
NE+1
1
Ne
b.
there is a natural number
for all x € Ce.
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