## Description

Assignment 9 of Math 5302

1. a) Write down all possible σ-algebras on X = {1, 2, 3, 4} which contains the element {1}.

b) Let A = {∅, {1}, {2, 3}, X} and B = {∅, {2}, {1, 3}, X}.

Find A ∪ B. Is A ∪ B a σ-algebra?

c) Find A ∩ B. Is A ∪ B a σ-algebra?

2. Prove that the intersection of any family of σ-algebras on X is a σ-algebra.

3. Prove that if N is a null set in R

n

, then there exists a Borel null set N0

such that N0 ⊆ N. Prove that

N0 may be chosen to be a “Gδ” set, a countable intersection of open sets.

4. Prove Property MF2: If f : X → R is M-measurable, and f 6= 0, then 1

f

is M-measurable.

5. Let E ⊆ R be a set which is not Lebesgue measurable. Let

f(x) =

e

x

if x ∈ E;

−e

x

if x ∈ E

c

.

(a) Prove that f is not Lebesgue measurable.

(b) Prove that for all t, f

−1

({t}) is Lebesgue measurable.

6. Let f : R → R be differentiable. Prove that the derivative f

0

is Borel measurable. (Be careful that f

0

may not be continuous.)

1