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2.1 Example Sheet 2: questions

Example 1.3 (Product $\sigma$-algebras). Of great importance for probability theory are product $\sigma$-algebras. Let $\mathcal{I}$ be an arbitrary index set, and for each $i \in \mathcal{I}$ let $\left(X_i, \mathcal{A}i\right)$ be a measurable space. The Cartesian product space $X=\prod{i \in \mathcal{I}} X_i$ is the space of all functions $x: \mathcal{I} \rightarrow \bigcup_{i \in \mathcal{I}} X_i$ such that $x(i) \in X_i$ for each $i$. Alternate notation for $x(i)$ is $x_i$. Coordinate projection maps on $X$ are defined by $f_i(x)=x_i$, in other words $f_i$ maps $X$ onto $X_i$ by extracting the $i$-coordinate of the $\mathcal{I}$-tuple $x$. The product $\sigma$-algebra $\bigotimes_{i \in \mathcal{I}} \mathcal{A}_i$ is by definition the $\sigma$-algebra generated by the coordinate projections $\left{f_i\right.$ : $i \in \mathcal{I}}$

1.1.2. Measures. Let us move on to discuss the second fundamental ingredient of integration. Let $(X, \mathcal{A})$ be a measurable space. A measure is a function $\mu: \mathcal{A} \rightarrow[0, \infty]$ that satisfies these properties:
(i) $\mu(\emptyset)=0$.
(ii) If $\left{A_i\right}$ is a sequence of sets in $\mathcal{A}$ such that $A_i \cap A_j=\emptyset$ for all $i \neq j$ (pairwise disjoint is the term), then
$$\mu\left(\bigcup_i A_i\right)=\sum_i \mu\left(A_i\right) .$$
Property (ii) is called countable additivity. It goes together with the fact that $\sigma$-algebras are closed under countable unions, so there is no issue about whether the union $\bigcup_i A_i$ is a member of $\mathcal{A}$. The triple $(X, \mathcal{A}, \mu)$ is called a measure space.

If $\mu(X)<\infty$ then $\mu$ is a finite measure. If $\mu(X)=1$ then $\mu$ is a probability measure. Infinite measures arise naturally. The ones we encounter satisfy a condition called $\sigma$-finiteness: $\mu$ is $\sigma$-finite if there exists a sequence of measurable sets $\left{V_i\right}$ such that $X=\bigcup V_i$ and $\mu\left(V_i\right)<\infty$ for all $i$. A measure defined on a Borel $\sigma$-algebra is called a Borel measure.

Example 1.26. Let $A$ be an event such that $0<P(A)<1$, and $\mathcal{A}=$ $\left{\emptyset, \Omega, A, A^c\right}$. Then
$$E(X \mid \mathcal{A})(\omega)=\frac{E\left(\mathbf{1}A X\right)}{P(A)} \cdot \mathbf{1}_A(\omega)+\frac{E\left(\mathbf{1}{\left.A^c X\right)}\right.}{P\left(A^c\right)} \cdot \mathbf{1}{A^c}(\omega)$$ Let us check (1.32) for $A$. Let $Y$ denote the right-hand-side of (1.35). Then \begin{aligned} \int_A Y d P & =\int_A \frac{E\left(\mathbf{1}_A X\right)}{P(A)} \mathbf{1}_A(\omega) P(d \omega)=\frac{E\left(\mathbf{1}_A X\right)}{P(A)} \int_A \mathbf{1}_A(\omega) P(d \omega) \ & =E\left(\mathbf{1}_A X\right)=\int_A X d P . \end{aligned} A similar calculation checks $\int{A^c} Y d P=\int_{A^c} X d P$, and adding these together gives the integral over $\Omega$. $\emptyset$ is of course trivial, since any integral over $\emptyset$ equals zero. See Exercise 1.15 for a generalization of this.

Here is a concrete case. Let $X \sim \operatorname{Exp}(\lambda)$, and suppose we are allowed to know whether $X \leq c$ or $X>c$. What is our updated expectation for $X$ ? To model this, let us take $(\Omega, \mathcal{F}, P)=\left(\mathbf{R}{+}, \mathcal{B}{\mathbf{R}_{+}}, \mu\right)$ with $\mu(d x)=\lambda e^{-\lambda x} d x$, the identity random variable $X(\omega)=\omega$, and the sub- $\sigma-$ algebra $\mathcal{A}={\Omega, \emptyset,[0, c],(c, \infty)}$. For example, for any $\omega \in(c, \infty)$, we would compute
\begin{aligned} E(X \mid \mathcal{A})(\omega) & =\frac{1}{\mu(c, \infty)} \int_c^{\infty} x \lambda e^{-\lambda x} d x=e^{\lambda c}\left(c e^{-\lambda c}+\lambda^{-1} e^{-\lambda c}\right) \ & =c+\lambda^{-1} \end{aligned}
You probably knew the answer from the memoryless property of the exponential distribution? If not, see Exercise 1.13. Exercise 1.16 asks you to complete this example.

The next theorem lists the main properties of the conditional expectation. Equalities and inequalities concerning conditional expectations are almost sure statements, although we did not indicate this below, because the conditional expectation is defined only up to null sets.

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