Lecture 4 (Part 6): Lebesgue dominated convergence theorem

This course is about the mathematical foundations of randomness. Most advanced topics in stochastics and statistics rely on probability theory. The basic constructions are identical to measure theory, but there are a number of distinctly probabilistic features such as independence, notions of convergence of random variables, information contained in a sigma-algebra, conditional expectation, characteristic functions and generating functions, laws of large numbers and central limit theorems, etc. @RUeamHK0X6#

[mikelindenstrauss.]

The statement of the Lebesgue's dominated convergence theorem is much more general than what you wrote. It says that if there is a sequence of measurable functions {f_n} dominated by an integrable function almost everywhere in a complete measure space converging to another function f almost everywhere then the integral of f_n converges to the integral of f.