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 present eight proofs of where is a standard normal random variable. This fact is more or less equivalent to saying that the Gaussian is its own Fourier transform. We start with proofs that are rather elementary, and move on to more sophisticated proofs.
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:
We present a polished version of Borel’s proof based on Cantor’s theory of sets. More specifically, we use transfinite recursion and the fact that any strictly decreasing sequence in a well-ordered set is finite in length.