Measurement theory and integration

Integration theory can naturally be approached from two very different angles. The first approach is a functional presentation, which first defines measures as elements of the dual of continuous functions with compact support. It then extends this notion to the larger class of integrable functions. The second approach, which we will present briefly in this article, is based directly on the notion of positive measure. It is this approach that allows the natural introduction of probabilities as positive measures of mass 1.

It is therefore important to know the foundations of measure theory - tribes, measurable functions, positive measures - in order to understand the probabilistic model. We'll also see that the Lebesgue measure is just a special case of a positive measure. Integration theory consists mainly in constructing the Lebesgue integral. It is based on a number of fundamental theorems (Beppo-Levi, Fatou, Lebesgue), the notion of product measure and Fubini's theorem.