The simple case of almost sure representations

The almost-sure representation theorem in probability says that if \(X_n\stackrel{d}{\to}X\) we can find a sequence \(\tilde X_n, \tilde X\) with the same distributions as \(X_n,X\), possibly on a different probability space, so that \(\tilde X_n\stackrel{a.s.}{\to}\tilde X\) (where the a.s. is with respect to the distribution of \(X\)). General versions of this theorem are a bit tricky to prove, and extremely general versions are very tricky to prove. There’s an extremely simple version that gives the idea, though.