Lubesgue Measure

Lebesgue measure
comparing different sets
Henri Lebesgue
1875-1941

Imagine you have a line that goes on forever. The Lebesgue measure allows us to measure how long a specific part of that line is. For example, if we want to measure the length of a line segment from point A to point B, we can use the Lebesgue measure to find out how many units long that segment is.

The Lebesgue measure is a way to measure the size of different shapes or sets of numbers. It helps us understand how big or small something is. For example, it can tell us the length of a line, the area of a shape on a flat surface, or the volume of a solid object. It was invented by a mathematician named Henri Lebesgue a long time ago.

In three dimensions, the Lebesgue measure helps us measure the volume of solid objects like cubes, spheres, or cylinders. It tells us how much space those objects occupy in three-dimensional space.

A Lebesgue measure is a useful tool in mathematics because it allows us to compare different sets of numbers and understand their sizes. It helps us quantify and make sense of the world around us by giving us a way to measure and describe different shapes and sizes. Example: measuring the height of the trees in a forest.

It is especially useful in a branch of math called real analysis and is used to define something called the Lubesgue integration which is the fixed version of Riemann integral. Let’s explain it with the graphs to make it simple. We have two mountains to calculate their 2D side coverage area.

In Reinmann integral, each segment is sliced into 18 gray 1-inch bars equally on the X axis to calculate. On the other hand, Lebesgue integral slices each segment into half-inch 7 different gray tones on the Y axis. Up to 12 calculations less. What if it was 1 billion bars? With a similar graph, Reinmann would have 1 billion, and Lebesgue still has up to 12 calculations.