A short story of measure theory

The Riemann integral is founded on the following idea: divide up the domain of a function f : [a, b] −→ R into subintervals, estimate f from above and below on each interval, and approximate the integral of f by the upper and lower sums—the summation of the widths of the intervals times the upper and lower estimates on each. The limit over partitions of [a, b] of these two approximations, should they exist and agree, is declared to be the integral of f . Sadly, this definition of the integral lacks some desirable properties. In particular, the space of absolutely integrable functions is not complete—a sequence of functions which is Cauchy in the norm ‖f‖1 = ∫ b a |f(x)| dx need not converge to a Riemann-integrable function. As a remedy to such deficiencies, the Lebesgue integral is founded on a different idea: namely, divide up the range of f into subintervals, and approximate the integral by the summation of the lower endpoints times the volume, or measure, of the interval’s preimage under f . To make this idea precise, we require