---- > [!definition] Definition. ([[product of σ-algebras]]) > Suppose $(X_{i})_{i \in I}$ is an indexed family of sets and $\Sigma_{i}$ is a $\sigma$-algebra on $X_{i}$. The [[σ-algebra generated by a set collection|initial σ-algebra]] on $\prod_{i \in I}^{}X_{i}$ wrt the [[categorical product|coordinate projections]] $\prod_{i \in I}^{}X_{i} \to X_{i}$ is denoted by $\bigotimes_{i \in I} \Sigma_{i}$ and called the **product of the $\sigma$-algebras $\Sigma_{i}$**. Explicitly, $\bigotimes_{i \in I} \Sigma_{i}$ is the [[σ-algebra]] [[σ-algebra generated by a set collection|generated by]] the collection of **cylinder sets**[^1] $\left\{ \prod_{i \in I} E_{i} : E_{i} \in \Sigma_{i} \text{ with } E_{i}=X_{i} \text{ for all but finitely many } i \in I \right\}.$ ^definition > [!definition] Definition. ([[product of σ-algebras|marginal measure]]) > By construction, each coordinate [[projection]] $\prod_{i \in I}^{} X_{i}\to X_{i}$ is [[measurable function|measurable]], and thus [[pushforward measure|pushes forward]] any [[measure]] $\mu$ on $\left( \prod_{i \in I}^{}X_{i}, \bigotimes_{i \in I} \Sigma_{i} \right)$ to a [[measure]] $\mu_{i}:=\mu_{X_{i}}$ on $X_{i}$. We call $\mu_{i}$ the **$i$th marginal measure of $\mu$**. ^definition [^1]: The justification is as in the exactly analogous notion in [[product topology]] (more generally, [[initial topology]]). > [!definition] Definition. ([[product of σ-algebras|finite products]]) For easy reference, we record the terminology for finite products below. > > Suppose $(X, \Sigma)$ and $(Y, \mathcal{T})$ are [[σ-algebra|measurable spaces]]. Then the **product of $\sigma$-algebras** $\Sigma \otimes \mathcal{T}$ is defined to be the [[σ-algebra]] [[σ-algebra generated by a set collection|generated by]] the set collection $\mathscr{A}:=\{ A \times B: A \in \Sigma, B \in \mathcal{T} \}$. The generating sets $A \times B$ are called **measurable rectangles in $\Sigma \otimes \mathcal{T}$**.[^2] > ^definition > [!basicproperties] > - $\mathcal{B}(X \times Y) \supset \mathcal{B}(X) \otimes \mathcal{B}(Y)$ for [[topological space|topological spaces]] $X,Y$. A sufficient condition for equality is $X$ or $Y$ being [[second-countable space|second-countable]]. (See [[products and Borel σ-algebras commute for second-countable spaces]].) > > - *(Cross section of measurable sets is measurable)* If $E \in \Sigma \otimes \mathcal{T} \subset 2^{(X \times Y)}$, then $[E]_{a}:= \{ y \in Y: (a,y) \in E \} \in \mathcal{T} \text{ for all } a \in X$ > and $[E]^{b}:= \{ x \in X : (x, b) \in E \} \in \Sigma \text{ for all }b \in Y.$ > > > [!proof]- Proof. > > > > Let $\mathcal{E}$ denote the collection of subsets $E$ of $X \times Y$ for which the result holds. The goal is to show $\Sigma \otimes \mathcal{T} \subset \mathcal{E}$, by showing that $\mathcal{E}$ contains a [[σ-algebra generated by a set collection|generating set]] and is stable under complementation and countable union (i.e., is a [[σ-algebra]]). > > > > If $E$ is a [[rectangle]] $E=A \times B$ with $A \in \Sigma$, $B \in \mathcal{T}$, then $[E]_{a}=B \in \mathcal{T}$ and $[E]^{b}=A \in \Sigma$. So $A \times B \in \mathcal{E}$ for all $A \in \Sigma$ and $B \in \mathcal{T}$. > > > > $\mathcal{E}$ is closed under complementation because if $E \in \mathcal{E}$ then $[X \times Y-E]_{a} = \{ y \in Y : (a,y) \not \in E \}= Y - [E]_{a} \in \mathcal{T}$ > > and similarly $[X \times Y - E]^{b}=X - [E]^{b} \in \Sigma.$ > > $\mathcal{E}$ is closed under countable union because if $E_{1},E_{2},\dots \in \mathcal{E}$ then $[E_{1} \cup E_{2} \cup \dots]_{a}= \{ y \in Y: (a, y) \in E_{1} \text{ or }E_{2} \text{ or }\dots\}=[E_{1}]_{a} \cup [E_{2}]_{a} \cup \dots $ > > and similarly for $[\cdot]^{b}$. > > > - *(Cross section of [[measurable function|measurable functions]] is measurable)* > > If $f:(X, \Sigma) \times (Y, \mathcal{T}) \to (Z, \mathcal{S})$ is a [[measurable function]], then so are its **cross sections** $[f]_{a}: Y \to Z$, $[f]^{b}:X \to Z$ for all $a \in X$, $b \in Y$, where $[f]_{a}(y):= f(a,y) \text{ for }y \in Y \text{ and }[f]^{b}(x)=f(x, b) \text{ for }x \in X.$ > > > [!proof]- Proof. > > Suppose $D \in \mathcal{S}$ is a measurable subset of $Z$. We claim $[f]_{a} ^{-1}(D)=[f ^{-1} D]_{a}$, where the latter is a measurable subset because $f$ is measurable and cross sections of measurable sets are measurable. Indeed: $\begin{align} > > y \in [f]_{a} ^{-1} (D) & \iff [f]_{a}(y) \in D \\ > > & \iff f(a, y) \in D \\ > > & \iff (a, y) \in f ^{-1} (D) \\ > > & \iff y \in [f ^{-1} (D)]_{a}. > > \end{align}$ > > The same idea goes through for $[f]^{b}$. > ---- #### [^2]: Extend to the case of finite products by induction. Despite the notation, this notion has scant relation to the [[tensor product of algebras]] familiar from [[ring|ring theory]] and (more generally) the theory of [[abelian category|abelian]] [[category|categories]]. Indeed, linearity is not under consideration at all here. A more appropriate comparison is to the [[product topology]] definition. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```