# Probability Theory: Homework 1

Exercises, due Wed Sept 3.

From Jacod and Protter, problems 2.6, 2.7, 2.10 and 2.14.

2.6 Let $\mathcal{A}$ be a $\sigma$-algebra of subsets of $\Omega$ and let $B \in \mathcal{A}$. Show that $\mathcal{F} := \{A \cap B \, : \, A \in \mathcal{A}\}$ is a $\sigma$-algebra of subsets of $B$.  Is this still true if $B$ is a subset of $\Omega$ that is not a member of $\mathcal{A}$?

2.7  Let $f$ be a function mapping $\Omega$ to another space $E$ with a $\sigma$-algebra $\mathcal{E}$. Let $\mathcal{A} := \{A \subset \Omega \, : \, \text{there exists } B \in \mathcal{E} \text{ with } A = f^{-1}(B)\}$. Show that $\mathcal{A}$ is a $\sigma$-algebra on $\Omega$.

2.10 Using the definition of a probability measure, show that if $A, \, B\, \in \mathcal{A}$, then $\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)$.

2.14 Suppose that $\mathbb{P}(A) = \frac{3}{4}$ and $\mathbb{P}(B) = \frac{1}{3}$. Show that $\frac{1}{12} \leq \mathbb{P}(A \cap B) \leq \frac{1}{3}$.