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Assignment 9 of Math 5302

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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

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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.)
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