# The Unapologetic Mathematician

## Product Measures

We continue as yesterday, considering the two $\sigma$-finite measure spaces $(X,\mathcal{S},\mu)$ and $(Y,\mathcal{T},\nu)$, and the product measure space $(X\times Y,\mathcal{S}\times\mathcal{T})$.

Last time we too a measurable set $E\subseteq X\times Y$ and defined the functions $f(x)=\nu(E_x)$ and $g(y)=\mu(E^y)$. We also showed that

$\displaystyle\int f\,d\mu=\int g\,d\nu$

That is, for every measurable $E$ we can define the real number

$\displaystyle\lambda(E)=\int\nu(E_x)\,d\mu=\int\mu(E^y)\,d\nu$

I say that this function $\lambda$ is itself a $\sigma$-finite measure, and that for any measurable rectangle $A\times B$ we have $\lambda(A\times B)=\mu(A)\nu(B)$. Since measurable rectangles generate the $\sigma$-ring $\mathcal{S}\times\mathcal{T}$, this latter condition specifies $\lambda$ uniquely.

To see that $\lambda$ is a measure, we must show that it is countably additive. If $\{E_n\}$ is a sequence of disjoint sets then we calculate

\displaystyle\begin{aligned}\lambda\left(\biguplus\limits_{n=1}^\infty E_n\right)&=\int\nu\left(\left(\biguplus\limits_{n=1}^\infty E_n\right)_x\right)\,d\mu\\&=\int\nu\left(\biguplus\limits_{n=1}^\infty(E_n)_x\right)\,d\mu\\&=\int\sum\limits_{n=1}^\infty\nu\left((E_n)_x\right)\,d\mu\\&=\sum\limits_{n=1}^\infty\int\nu\left((E_n)_x\right)\,d\mu\\&=\sum\limits_{n=1}^\infty\lambda(E_n)\end{aligned}

where we have used the monotone convergence theorem to exchange the sum and the integral.

We verify the $\sigma$-finiteness of $\lambda$ by covering each measurable set $E$ by countably many measurable rectangles with finite-measure sides. Since the sides’ measures are finite, the measure of the rectangle itself is the product of two finite numbers, and is thus finite.

We call the measure $\lambda$ the “product” of the measures $\mu$ and $\nu$, and we write $\lambda=\mu\times\nu$. We thus have a $\sigma$-finite measure space $(X\times Y,\mathcal{S}\times\mathcal{T},\mu\times\nu)$ that we call the “cartesian product” of the spaces $(X,\mathcal{S},\mu)$ and $(Y,\mathcal{T},\nu)$.

July 23, 2010 Posted by | Analysis, Measure Theory | 6 Comments