Product measure

The product measure ${⨂}_n{\mu }_n$ is characterized on measurable rectangles depending on finitely many coordinates by the product of their component measures.

The product sigma-algebra on $X_1\times X_2$ is generated by the measurable rectangles $A_1\times A_2$. It is the smallest sigma-algebra making both coordinate projections measurable.

If a Dynkin system contains a pi-system, then it contains the sigma-algebra generated by that pi-system. This proves uniqueness of finite measures from agreement on a generating pi-system.

For a probability law $\nu$ on $E$, take the sequence space $E^{\mathbb{N}}$ with product measure ${\nu }^{\otimes \mathbb{N}}$. The coordinate maps are an independent and identically distributed sequence with common law $\nu$, and they generate the product sigma-algebra.