Radon–Nikodym Theorem with Implications in Probability Theory

This essay focuses on the Radon-Nikodym Theorem, a cornerstone result in measure theory that relates two measures defined on the same measurable space when one is absolutely continuous with respect to the other. To understand the behaviour of the Radon-Nikodym Theorem, we define an integral with respect to an arbitrary measure space, develop essential concepts in measure theory, and present critical results such as the Monotone Convergence Theorem. An essential application of the Radon-Nikodym Theorem is in probability theory, where it provides a rigorous foundation for defining conditional probability and expectation.