Products of Measures

Abstract

Just before Lemma 3.1.1, I introduced the product $(E_1\times E_2,\mathcal B_1\times \mathcal B_2)$ of two measurable spaces $(E_1,\mathcal B_1)$ and $(E_2,\mathcal B_2)$. In the present chapter I will show that if $\mu _i$, $i\in \{1,2\}$, is a measure on $(E_i,\mathcal B_i)$, then, under reasonable conditions, there is a unique measure $\nu =\mu _1\times \mu _2$ on $(E_1\times E_2,\mathcal B_1\times \mathcal B_2)$ with the property that $\nu ({\varGamma }_{\!1}\times {\varGamma }_{\!2})=\mu _1({\varGamma }_{\!1})\,\mu _2({\varGamma }_{\!2})$ for all ${\varGamma }_{\!i}\in \mathcal B_i$. In addition, I will derive several important properties relating integrals with respect to $\mu _1\times \mu _2$ to iterated integrals with respect to $\mu _1$ and $\mu _2$. Finally, in § 4.2, I will apply these properties to derive the isodiametric inequality.