Infinite product measure

Remarks

Such a measure exists for every family of probability spaces, without topological assumptions, and it is unique because the cylinder sets form a π-system that generates $\Sigma$. For countable $I$, existence also follows from the Ionescu-Tulcea theorem with kernels that do not depend on the past.

Example: let $X_n = \set{0, 1}$ and $\mu_n(\set{0}) = \mu_n(\set{1}) = \tfrac12$ for $n = 1, 2, 3, \ldots$ The product measure on the space of sequences $(x_1, x_2, \ldots)$ with $x_n \in \set{0, 1}$ is the law of a sequence of fair coin tosses. The set of sequences starting with $0, 1$ is a cylinder set with $J = \set{1, 2}$ and measure $\tfrac12 \cdot \tfrac12 = \tfrac14$.

Non-example: on the same space, the Dirac measure at the constant sequence $(0, 0, 0, \ldots)$ is a probability measure. It gives the cylinder set $\set{x : x_1 = 0}$ measure $1$ instead of $\tfrac12$, so it is not the product of the fair coin measures.

Under the product measure, the coordinate maps $\pi_i$ are independent random variables, and $\pi_i$ has law $\mu_i$.