Overview of Measure Theory

The ideas presented in this theory are fairly general, but their utility will not be immediately visible. However, a general note is that we are trying to quantify how “small” or how “large” a set is. Recall that if a bounded function has only finitely many discontinuities, it is still Riemann-integrable. The Dirichlet function was an example of a function with uncountably many discontinuities, and it failed to be Riemann-integrable. Is there a notion of “smallness” such that if a function is bounded except for a small set, then we can compute its integral? For example, we could say a set is “small” on the real line if it has at most countably infinitely many elements. Unfortunately, this is not enough to deal with the integral of the Dirichlet function. So we need a notion adequate to deal with the integral of functions such as the Dirichlet function.