Skorokhod's representation theorem

In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a sequence of random variables defined on a common probability space. It is named for the Ukrainian mathematician A.V. Skorokhod.

Statement of the theorem
Let $$(\mu_{n})_{n = 1}^{\infty}$$ be a sequence of probability measures on a topological space $$S$$; suppose that $$\mu_{n}$$ converges weakly to some probability measure $$\mu$$ on $$S$$ as $$n \to \infty$$. Suppose also that the support of $$\mu$$ is separable. Then there exist random varables $$X_{n}, X$$ defined on a common probability space $$(\Omega, \mathcal{F}, \mathbb{P})$$ such that
 * $$(X_{n})_{*} (\mathbb{P}) = \mu_{n}$$ (i.e. $$\mu_{n}$$ is the distribution/law of $$X_{n}$$);
 * $$X_{*} (\mathbb{P}) = \mu$$ (i.e. $$\mu$$ is the distribution/law of $$X$$); and
 * $$X_{n} (\omega) \to X (\omega)$$ as $$n \to \infty$$ for every $$\omega \in \Omega$$.