Return to Revisions

There's even more. If your set can be considered as a product of probability spaces, its measure will have some good properties. This is a theorem due to Alexandra Ionescu Tulcea: given an infinite (with arbitrary cardinality) family of probability spaces $$\{(\Omega_n, \mathscr{B}_n, \mu_n)\}_{n \in I}$$ if we call $$\pi_m : \prod_{n \in I} \Omega_n \rightarrow \Omega_m$$ the canonical projection, then it's possible to give $\Omega \mathrel{\mathop{:}=} \prod_{n \in I} \Omega_n$ the structure of a probability space with measurable sets $$\mathscr{B} = \bigotimes_{n \in I} \mathscr{B}_n$$ and probability measure $\mu : \mathscr{B} \rightarrow \mathbb{R}$ such that $$\forall \{n_1, \ldots, n_t\} \subset I \ \ \forall (A_{n_1}, \ldots, A_{n_t}) \in \prod_{i = 1}^t \mathscr{B}_{n_i}$$ $$\mu\left(\pi_{n_1}^{-1}(A_{n_1}) \cap \ldots \cap \pi_{n_t}^{-1}(A_{n_t})\right) = \mu_{i_1}(A_{i_1}) \ldots \mu_{i_t}(A_{i_t})$$