Lebesgue integral

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the X axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions.

The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the Riemann integral. The Lebesgue integral also has generally better analytical properties. For instance, under mild conditions, it is possible to exchange limits with Lebesgue integration, while the conditions for doing this with a Riemann integral are comparatively restrictive.

The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue, or the specific case of integration of a function defined on a sub-domain of the real line with respect to the Lebesgue measure.

Introduction

The integral of a positive function can be thought of as the area under its graph. For many functions, this idea works well, but some functions, like the Dirichlet function, are harder to handle. Mathematicians needed a better way to understand these areas.

The Riemann integral, created by Bernhard Riemann, was a good start. It uses vertical rectangles to estimate area. But it has limits, especially when dealing with limits of functions. The Lebesgue integral, named after Henri Lebesgue, uses a different approach. Instead of vertical rectangles, it uses horizontal slabs. This makes it better for many types of functions, including the Dirichlet function.

In simple terms, the Lebesgue integral looks at how values of the function are grouped, which allows for more flexibility and better handling of complex functions.

Definition

The Lebesgue integral is a way to calculate the area under a curve, even for more complicated functions than the usual method (called the Riemann integral). It was developed by a French mathematician named Henri Lebesgue.

One way to build the Lebesgue integral is by using simple functions. These are functions that take only a few different values. By breaking down a complicated function into simpler pieces, we can calculate its integral step by step. This method works for many types of functions and helps us understand integrals better.

Main article: Layer cake representation

Example

The indicator function of the rational numbers, shown as 1_Q, cannot be integrated using the Riemann method. This is because, no matter how you divide the space between 0 and 1, you'll always find both rational and irrational numbers in every piece.

However, using the Lebesgue method, this function can be integrated. The Lebesgue method looks at how much space the rational numbers take up, and since there are only countably many rational numbers, their total space is zero. So, the Lebesgue integral of this function over the interval from 0 to 1 equals zero.

Domain of integration

In Lebesgue integration, the area we calculate doesn’t depend on a direction, unlike in basic calculus where integrating from point b to a gives the opposite result of integrating from a to b. Instead, Lebesgue integration focuses on measuring parts of space using something called a measure. This method allows us to calculate over any selected region, not just intervals on a line. This approach connects to more advanced topics like integrating over shapes in higher dimensions and a theory called homological integration, developed by mathematicians Georges de Rham and Hassler Whitney.

Main article: Differential form § Relation with measures
Main articles: Homological integration, Georges de Rham, Hassler Whitney

Limitations of the Riemann integral

The Riemann integral, while useful, has some problems when dealing with certain mathematical tasks. One issue arises when we try to switch the order of taking limits and integrating functions. This can be tricky and doesn't always work well with the Riemann approach.

The Riemann integral also struggles with some special functions and areas. For example, it works best on straight, simple shapes and with functions that behave nicely, but it can fail with more complex functions or unusual areas. While there are ways to extend the Riemann integral to handle some of these cases, many functions that can be handled by the Lebesgue integral are still too difficult for the Riemann method. Additionally, the Riemann integral is closely tied to the natural order of numbers on a line, which limits its use in more complex mathematical spaces.

Main article: Monotone convergence theorem

Basic theorems of the Lebesgue integral

Two functions can be considered the same if they are equal almost everywhere, meaning they differ only on a very small set of points. The Lebesgue integral has important properties that make it useful in mathematics.

Some key ideas include linearity, where the integral of a sum of functions equals the sum of their integrals, and monotonicity, where the integral of a smaller function is less than or equal to the integral of a larger function. Important theorems like the monotone convergence theorem and Fatou's lemma help us understand how integrals behave when we take limits of sequences of functions.

Alternative formulations

There are different ways to understand the Lebesgue integral without needing all the complex ideas of measure theory. One method is called the Daniell integral.

Another way uses ideas from functional analysis. We start with simple continuous functions and build up to more complicated ones. This process helps us define the Lebesgue integral in a clear and logical way.

Limitations of Lebesgue integral

The Lebesgue integral helps us understand how to add up areas under curves, even for very complex shapes. However, not every function can be handled this way. For example, the sinc function — which looks like a wave that gets smaller and smaller as it moves away from the center — cannot be fully integrated using the Lebesgue method because the total area would become infinite.

Even though the Lebesgue method doesn’t work for this function, there is another way to find a meaningful result called an improper integral. This method gives a finite answer for the sinc function, which is useful in many areas of mathematics.