Simple Proofs of the Kolmogorov Extension Theorem and Prokhorov’s Theorem

We provide simple proofs of the Kolmogorov extension theorem and Prokhorov’s 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.

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