Sequences of Functions and Convergence Theorems

Abstract

The convergence theorems we start with yield sufficient conditions that the integral can be interchanged with pointwise convergence of functions. These theorems clearly require continuity of the set function. First the Monotone Convergence Theorem derives from the special case proved in Chapter 1. It is valid very generally for monotone set functions. The other important theorem, Lebesgue’s Dominated Convergence Theorem, is first proved for the classical case of measures and later on it is generalized for subadditive set functions in applying the classical case with Lebesgue measure on the distribution functions. For this purpose we weaken pointwise convergence to stochastic convergence and further to convergence in distribution. These different types of convergence are important, too, in probability theory and statistics. By the way we get some information on measurability of the limit function even if the set function is not continuous.