Week 1 Worksheet

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1. Suppose that $\Xi : \mathcal{P}(\R) \to [0,\infty]$ is countably additive. Carefully prove that \[ \Xi(A \cup B) = \Xi(A) + \Xi(B) \] whenever $A,B$ are disjoint subsets of $X$.
2. Prove $\Lambda(A - t) = \Lambda(A)$ 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 $t$.
3. Prove $\Lambda( [0,1] ) = 1$.
4. Carefully verify that $\{ \emptyset, X \}$ is a $\sigma$-algebra on $X$.
5. Fix a measurable space $(X,\B)$. Given a sequence $B_1,B_2,B_3,\dots$ in $\B$ define \[ \begin{aligned} \limsup_n B_n &= \bigcap_{N \in \N} \bigcup_{j \ge N} B_j \\ \liminf_n B_n &= \bigcup_{N \in \N} \bigcap_{j \ge N} B_j \end{aligned} \]
   1. Write sentences describing what it means for a point to belong to the above sets.
   2. Verify that $\limsup\limits_n B_n \supset \liminf\limits_n B_n$.
   3. Verify that $\limsup\limits_n B_n$ and $\liminf\limits_n B_n$ are measurable.
6. 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.
7. 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 $\R \setminus \Q$ 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 $B \symdiff U$ is meagre.
   3. Prove that the collection $\baire(\R)$ of Baire subsets of $\R$ is a $\sigma$-algebra.
   4. Prove that every Borel subset of $\R$ is a Baire subset of $\R$.
8. 1. How many Borel sets are there?
   2. Is every subset of the middle-thirds Cantor set a Borel set?