Math 524
Homework due 12/04/02
Reading: Sections 4, 5 and 6, Chapter 2 in Folland.
Problems from Folland:
Chapter 2, Section 3: problem 23.
Chapter 2, Section 4: problems 33, 34, 36, 40, 41, 44.
Hint for problem 36: see problems 3 and 5 in Chapter 2, Section 1.
∗Problem: Let (X, M, µ) be a measure space. Assume µ(X)<∞. Let {fn}n≥1be a
sequence in L1(µ) satisfying:
1. fn→f µ −a.e in X
2. Given > 0 there is δ > 0 so that
sup
n≥1ZE
|fn|dµ < , whenever µ(E)< δ.
Show that f∈L1(µ) and that fn→fin L1(µ), i.e.
lim
n→∞ Z|fn−f|dµ = 0.
1