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Homework 3, Undergraduate Analysis
Refer to class notes for definitions of sets and functions in this homework assignment.
All functions and sets are measurable.
1. If f ≥0 and R
Ef dµ= 0, then f = 0 a.e. on E.
2. Prove the following basic properties for Lebesgue integrations:
(a) f(x)∈[a, b] for all x∈X,µ(X)<∞, then
aµ(X)≤ Z
X
f dµ ≤bµ(X).
(b) f ≤g ⇒R
Xf dµ ≤R
Xg dµ.
(c) ∀c∈R,R
Ecf dµ=cR
Ef dµ.
(d) R
Ef dµ =R
Xf KE dµ.
(e) f ∈ L(X)⇒f ∈ L(E) for all measurable subset E of X.
3. Rudin Chapter 11, Exercise 8.
4. Rudin Chapter 11, Exercise 12.
5. Rudin Chapter 11, Exercise 11.
6. Rudin Chapter 11, Exercise 16.
7. If f ≥ 0 integrable on X, then for all > 0, there exists E ∈ M such that µ(E)< ∞ and
Z
E
f dµ >
Z
X
f dµ−.
