---- > [!definition] Definition. ([[product measure]]) > Suppose $(X, \Sigma, \mu)$ and $(Y, \mathcal{T}, \nu)$ are [[finite measure|σ-finite]] [[measure|measure spaces]]. There exists a unique measure $\mu \times \nu$ on $(X \times Y, \Sigma \otimes \mathcal{T})$ with the property that $(\mu \times \nu)(E \times F)=\mu(E)\nu(F)$. Explicitly, we define [[product of σ-algebras|a]] [[measure]] $\mu \times \nu$ [[σ-algebra|on]] $(X \times Y, \Sigma \otimes \mathcal{T})$ via $E \mapsto \int _{X} \int _{Y} \, \chi_{E} (x, y) \, d\nu (y) \, d\mu(x) .$ > This iterated [[integral]] is well-defined: > - The inner integral equals $\nu([E]_{x})$, [[integral#^equivalence|which makes sense]] since $[E]_{x}$ is [[σ-algebra|measurable]] as the [[product of σ-algebras|cross section of a measurable set]]; > - The outer integral is of $x \mapsto \nu([E]_{x})$, and this makes sense by [[measure of cross section is a measurable function]]. > [!generalization] > - [[transition kernel|semidirect product of measures]] ^generalization > [!justification] > > We need to check that the product $\mu \times \nu$ of two measures is indeed a [[measure]]. Clearly $(\mu \times \nu)(\emptyset)=0$. Suppose $E_{1},E_{2},\dots$ is a disjoint [[sequence]] of sets in $\Sigma \otimes \mathcal{T}$. Then countable additivity holds because $\begin{align} > (\mu \times \nu)(\bigsqcup_{k=1}^{\infty} E_{k}) &= \int _{X} \nu ([\bigsqcup_{k=1}^{\infty} E_{k}]_{x}) \, d\mu (x) \\ > &= \int _{X} \nu ( \bigsqcup_{k=1}^{\infty} [E_{k}]_{x})\, d\mu (x) \\ > &= \int _{X} \sum_{k=1}^{\infty} \nu([E_{k}]_{x}) \, d\mu (x) \\ > &= \sum_{k=1}^{\infty} \int _{X} \nu ([E_{k}]_{x}) \, d\mu (x) \\ > &= \sum_{k=1}^{\infty} (\mu \times \nu) (E_{k}) > \end{align}$ > where the fourth equality follows from the [[monotone convergence theorem for nonnegative measurable functions|monotone convergence theorem]]. ---- #### ---- #### 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 > ```