We provide new simple proofs of the Kolmogorov extension theorem and Prokhorovs' theorem. The proof of the Kolmogorov extension theorem is based on the simple observation that $\mathbb{ R}$ and the product measurable space $\{ 0,1\}^\mathbb{ N}$ are Borel isomorphic. To show Prokhorov's theorem, we observe that we can assume that the underlying space is $\mathbb{ R}^\mathbb{ N}$. Then the proof of Prokhorov's theorem is a straightforward application of the Kolmogorov extension theorem we just proved.
Categories: math.PR