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

TL;DR

This paper presents new, simplified proofs for the Kolmogorov extension theorem and Prokhorov's theorem, leveraging Borel isomorphisms and the Kolmogorov extension theorem itself.

Contribution

It introduces straightforward proofs for two fundamental theorems in probability theory, simplifying their understanding and application.

Findings

Proof of Kolmogorov extension theorem using Borel isomorphism.

Prokhorov's theorem derived as a straightforward application of the new proof.

Simplified approach enhances accessibility of key probabilistic theorems.

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.