$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ > [!definition] > Let $(\Omega_\alpha, \FFF_\alpha)_{\alpha\in I}$ be a collection of measure spaces. The ==**product $\sigma$-field**== $\FFF = \bigotimes_{\alpha\in I} \FFF_\alpha$ is the minimal $\sigma$-field over $\Omega = \bigotimes_{\alpha \in I} \Omega_\alpha$ such that each projection map $\pi_\alpha: \Omega\to \Omega_{\alpha}$ is measurable; equivalently, it is the $\sigma$-algebra generated by > $ E_\alpha \times \prod_{\gamma \in I\setminus \{a\}} \Omega_\gamma$ > for all $\alpha$. >[!idea] >If this isn't reminiscent of the [[product topology]], you're actually blind. The main intuition is that a product space should be independent in each of its projections. We're sometimes interested in the sets $ \Omega_J = \prod_{\alpha \in J} \Omega_\alpha,\qquad \FFF_J = \bigotimes_{\alpha\in J} \FFF_\alpha$ which are the partial products over subsets $J\subset I$; this will be standard notation. >[!example] >It's useful to note that >$ \AAA_0 = \{(\pi_J)^{-1}(E_J) = E_J\times \Omega_{I\setminus J}: \abs{J} < \infty, E_J\subset F_J\}$ >is an algebra generating our product $\sigma$-algebra. # How we want the Product Measure to behave The main intuition is that a product space should be independent in each of its projections. > [!definition] Marginals > Let $J\subset I$ be finite, and let $\PP$ be a measure on $\Omega$. Then, the ==**finite-dimensional marginal**== (or ==**distribution**==) of $\PP$ on $J$ is the pushforward measure $\PP_J = (\pi_J)_\sharp (\PP)$ on $(\Omega_J, \FFF_J)$. Explicitly, > $ \PP_J(E) = \PP(E\times \Omega_{I\setminus J})$ > for all $E\subset F_J$. We want these finite-dimensional marginals to all be self-consistent; if $J'\subset J\subset I$ are finite subsets, then the marginal $\PP_{J'}$ should be the same whether we project from $I$ to $J'$ or through $J$.