Measure, integration and probability theory



Can you assign a surface area to every possible subset of the plane? Can you determine the integral of every possible function? Try to answer these questions, and you will open a Pandora’s box called measure theory. But this curious by-product of innocent questions is foundational for at least two large areas of mathematics: Partial Differential Equations and Probability Theory. Without measure theory, many Partial Differential Equations could not even be formulated, let alone be shown to be well-posed. Measure theory really is the rigorous model for modern probability theory, and is essential for understanding general real-valued random variables and stochastic processes.
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