Name and Student ID’s:
Homework 1, Undergraduate Analysis 1
1. Fill in the gaps in the construction of the non-measurable space in class. (The equivalence relation, and the partition of [0, 1) into translations of N .)
2. Prove that any intersection of a family of σ−algebras is still a σ−algebra.
3. Rudin Chapter 11, Exercise 5.
4. Rudin Chapter 11, Exercise 6.
5. Given a set X, a family of subsets R ⊂ P(X) is called a ring if it is closed under finite unions and complements:
• E, F ∈ R ⇒ E\F ∈ R.
• E1, . . . , En ∈ R ⇒ ∪ni=1Ei ∈ R.
A ring is called σ-ring if is closed under countable union:
• {Ei}∞i=1⊂ R ⇒ ∪∞i=1Ei ∈ R.
Prove that a σ-ring R is a σ-algebra if and only if X ∈ R.
6. Let M be an infinite σ−algebra,
(a) Prove that M has an infinite sequence of non-empty disjoint sets.
(b) Prove that M is uncountable.
7. Prove that an algebra A is a σ-algebra if it is closed under countable increasing unions:
If {Ei}∞i=1 ⊂ A and Ei ⊂ Ei+1 for all i, then ∪∞i=1Ei ∈ A.
