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 and the product measurable space are Borel isomorphic. To show Prokhorov’s theorem, we observe that we can assume that the underlying space is . Then the proof of Prokhorov’sContinue reading “Simple Proofs of the Kolmogorov Extension Theorem and Prokhorov’s Theorem”
We show that many integral convergence theorems are equivalent to the dominated convergence theorem, and that they are strictly weaker than the monotone convergence theorem. The precise sense of these assertions are given.
The Hahn-Kolmogorov theorem (which is sometimes called the Carathéodory extension theorem) is used to construct various kind of measures. We provide an intuitive proof of this theorem which avoids using Carathéodory’s (rather) unintuitive criterion for measurability.
We prove Hölder’s inequality using the so-called layer cake representation and the tensor power trick. The heart of the proof is the following one line: