MT Lesson 17: Product measures

In this section, we talk about product measures.

Suppose two measurable space $(\Omega_1, \mathcal{F_1}, \mu_1)$ and $(\Omega_2, \mathcal{F_2}, \mu_2)$, $\Omega_1\times\Omega_2=\left{(x,y):x\in \Omega_1, y\in \Omega_2\right}=\Omega$. Suppose $E_1\in \mathcal{F_1}$ and $E_2\in \mathcal{F_2}$, then rectangle $E_1\times E_2=\left{(x,y):x\in E_1, y\in E_2\right}$. We are now going to well-define $\mu(E_1\times E_2)$ as $\mu_1(E_1)\mu_2(E_2)$.

DEF (rectangles)

OBS $\mathcal{F_1}\times \mathcal{F_2}$ is a semi-algebra.

PROP $\mu$ is additive.

LEMMA If $A\in \mathcal{F_1}*\mathcal{F_2}=\mathcal{F}$

1. $\forall x\in \Omega_1$, $A_x\in \mathcal{F_2}$,
2. $\forall y\in \Omega_2$, $A^y\in \mathcal{F_1}$

In fact , $\mu$ is $\sigma$-additive.