TITLE: need reference for fact about products of measures
QUESTION [7 upvotes]: Need a reference (textbook or paper) for the following (probably well known) fact: 
Suppose $(X,M)$ is a measurable space and $\lambda$, $\nu$ are two different probability measures on $(X,M)$. Write $\lambda^\infty$, $\nu^\infty$ for the corresponding infinite product measures on the $(X^\infty ,M^\infty )$. Then 
$\lambda^\infty$ and $\nu^\infty$ are mutually singular.

REPLY [8 votes]: Use the strong law of large numbers.
Choose $B\in M$ so that $\lambda(B)\neq \mu(B)$, and consider the disjoint sets
$$\left\{x\in X^\infty: {1\over n}\sum_{j=1}^n 1_{[x_j\in B]}\to \lambda(B)\right\}\mbox{ and }\left\{x\in X^\infty: {1\over n}\sum_{j=1}^n 1_{[x_j\in B]}\to \mu(B)\right\}.$$