The Lantern Festival is a traditional Chinese festival. When I was a child, for the Lantern Festival, my parents would always go and buy paper specifically for making lanterns for me to make my own. In recent years, my elders have asked me to teach my nephews and nieces how to make lanterns, and I always have to estimate how many square metres of lantern paper to buy. But what if the lantern is an irregular shape, like a crooked oval or a lopsided rectangle? How can I calculate the area then?
Last year, I took the Real Analysis Course and learnt the Lebesgue integration. The story I told makes me understand the Lebesgue integral better. Unlike Riemann integration, it’s a mathematical tool that allows us to calculate the area under a curve or the volume of a region, even if the shape is irregular.
Remember the Riemann integral we learned in high school? The Riemann integral and the Lebesgue integral are two different methods. While they both have the same goal, they differ in their approach to dividing the domain of the function into smaller pieces, and in the types of functions they can integrate. The Riemann integral is based on dividing the domain of the function into a finite number of small intervals, and approximating the area under the curve by summing up the areas of rectangles that approximate the curve on each interval. However, the Lebesgue integral is based on dividing the domain of the function into infinitely many small intervals, and approximating the area under the curve by measuring the sizes of subsets of the domain rather than their widths. This allows the Lebesgue integral to integrate a wider range of functions, including functions that are not continuous, have discontinuities, or behave wildly.
The Lebesgue integral has many practical applications, including in signal processing, finance, and physics. For instance, the Fourier transform, which is a key tool in signal processing, involves integrating over infinite intervals, which requires the use of the Lebesgue integral.
The Lebesgue integral allowed mathematicians to integrate a wider range of functions than the earlier Riemann integral, which led to new insights and discoveries in analysis, measure theory, and other fields. Lebesgue integral reminds us that innovation and creativity can lead to breakthroughs and new tools can lead to new insights!
University of Adelaide