Measure Theory, An Introduction

Introduction

As you recall, Riemann developed a technique that determines the area under a curve, or the volume under a surface, provided the shape is smooth, or at least piecewise smooth. He took finer and finer approximations, and proved that they approached a limit, which had to be the area of the shape. A real number q, which is the integral of a region R, is the area of R by definition, or, if you prefer, q is the area that is implied by Euclid's axioms, because subregions (fitting inside R) have areas approaching q from below, and superregions (containing R) have areas that approach q from above. So everybody's happy, right?

Not quite. Let f be a function on [0,1] that is 0 on the irrationals and 1 on the rationals. This is not riemann integrable, yet most of the time f(x) = 0. It seems like it should have an integral of 0. Beyond this intuition, certain branches of mathematics, such as probability and fourier analysis, would benefit from a more general definition of integration. Thus measure theory is sometimes viewed as a means to an end, a foundation that is needed to prove other important theorems in mathematics.

Lebesgue (pronounced Lebeg) (biography) introduced a more general approach to integration, which is based on measurable sets and measurable functions. In the following pages, I will define sigma algebras, measurable sets, measurable functions, and finally, lebesgue integration.

Many thanks to Geon Oh, who helped me with this topic.