4.2 Product $\s$-Algebra
Definition 4.2.1 (Product $\s$-Algebra).label Let $\curl{(X_i,\cali{M}_i)}_{i\in I}$ be measurable spaces, then the product $\s$-algebra $\tprods{}{i\in I}\cali{M}_{i}$ on $\prods{}{i\in I}X_{i}$ is the $\s$-algebra generated by $\curl{\pi_i}_{i\in I}$.
Lemma 4.2.2.label Let $\curl{(X_i,\cali{M}_i)}_{i\in I}$ be measurable spaces and
\begin{align*}\cali{E}=\curl{\caps{}{j\in J}\pi_j\inv (E_j):J\suf I\text{ finite, }E_j\in\cali{M}_j}\end{align*}
then $\cali{E}$ is an elementary family, and $\s(\cali{E})=\tprods{}{i\in I}\cali{M}_{i}$
Proposition 4.2.3 ([Proposition 1.5, Fol99]).label Let $\curl{X_j}_{1}^{n}$ be topological spaces, $X=\prods{n}{j=1}X_{j}$, then:
- (1)
$\tprods{n}{j=1}\cali{B}_{X_j}\suf\cali{B}_{X}$.
- (2)
If $\curl{X_j}_{1}^{n}$ are separable, then $\tprods{n}{j=1}\cali{B}_{X_j}=\cali{B}_{X}$.
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