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The Integral That Riemann Couldn’t Solve
How Henri Lebesgue Changed the Meaning of Area and Opened the Door to Modern Mathematics
Most of us who encountered calculus at school or university were taught integration as a method of finding the area beneath a curve. Draw a graph, divide the area beneath it into a large number of narrow rectangles, calculate their areas and add them together. Make the rectangles narrower and narrower and the answer becomes increasingly accurate. It is an elegant idea, and for most of the functions encountered in ordinary mathematics it works extremely well.
This approach is closely associated with the German mathematician Bernhard Riemann, who developed a rigorous definition of integration during the nineteenth century. The Riemann integral became one of the foundations of mathematical analysis and remains the form of integration most students encounter first. There was, however, a problem. Mathematicians gradually discovered increasingly strange functions for which Riemann’s apparently straightforward method did not work.
Riemann’s Way of Looking at a Function
Imagine a graph with a curve running across it. Riemann essentially divides the horizontal axis into small sections. For each section, we construct a rectangle whose height is determined by the function. Adding the areas of…
