Week 1 Worksheet

Suppose that $Ξ : P (R) \to [0, \infty]$ is countably additive. Carefully prove that $Ξ (A \cup B) = Ξ (A) + Ξ (B)$ whenever $A, B$ are disjoint subsets of $X$ .
Prove $Λ (A - t) = Λ (A)$ for every $A \subset R$ and every $t \in R$ where $A - t = {s \in R : s + t \in A}$ is the translate of $A$ by $t$ .
Prove $Λ ([0, 1]) = 1$ .
Carefully verify that ${\emptyset, X}$ is a $σ$ -algebra on $X$ .
Fix a measurable space $(X, B)$ . Given a sequence $B_{1}, B_{2}, B_{3}, \dots$ in $B$ define $\begin{aligned} \underset{n}{lim sup} B_{n} & = ⋂_{N \in N} ⋃_{j \geq N} B_{j} \\ \underset{n}{lim inf} B_{n} & = ⋃_{N \in N} ⋂_{j \geq N} B_{j} \end{aligned}$
1. Write sentences describing what it means for a point to belong to the above sets.
2. Verify that $\underset{n}{lim sup} B_{n} \supset \underset{n}{lim inf} B_{n}$ .
3. Verify that $\underset{n}{lim sup} B_{n}$ and $\underset{n}{lim inf} B_{n}$ are measurable.
1. Write down a Borel subset of $R$ that is not open.
2. Write down a Borel subset of $R$ that is neither open nor closed.
3. Write down a Borel subset of $R$ that is neither a countable union of open sets nor a countable union of closed sets.
Say that $M \subset R$ is nowhere dense if its closure has empty interior. Say that $M \subset R$ is meagre if it is a countable union of nowhere dense sets.
1. Prove that countable sets are meager.
2. Prove that $R ∖ Q$ is not meager.
Say that $B \subset R$ is a Baire subset of $R$ if there is an open set $U \subset R$ such that $B △ U$ is meagre.
1. Prove that the collection $Baire (R)$ of Baire subsets of $R$ is a $σ$ -algebra.
2. Prove that every Borel subset of $R$ is a Baire subset of $R$ .
1. How many Borel sets are there?
2. Is every subset of the middle-thirds Cantor set a Borel set?