Samuel Lockman: Kolmogorov’s Theorem, Haar Functions and Brownian Motion — A Measure Theoretic Introduction

Abstract:

In this text, we seek to give a measure theoretic introduction to Brownian Motion and prove two of its main properties. Part of it is done through the continuity theorem of Kolmogorov, which in itself can be considered a central result. Furthermore, the paper also includes a construction of a Brownian motion via the Haar functions by first proving it is an orthonormal basis of \( L^2 [0, 1]\).