New simple proofs of the Kolmogorov extension theorem and Prokhorov's theorem

Abstract: 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.