Measurable Functions and Sets. Jensen’s Inequality. The Theorem of Egorov

Abstract

We introduce the general notion of a measurable function and a measurable set. Measurable functions are characterized as pointwise limits of finite valued functions. Jensen’s inequality for the integration of convex functions and Egorov’s theorem saying that an almost everywhere converging sequence of functions also converges almost uniformly, i.e. uniformly except on a set of arbitrarily small measure, are derived. We conclude this § with an introduction to the general theory of measures.