Lebesgue integral

In mathematics, the integral of a non-negative function can be thought of as the area between the graph of that function and the X axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, is a way to make this idea precise and to use it for more kinds of functions.

The Lebesgue integral is more general than the Riemann integral. It can handle functions that have sudden changes, which are harder for the Riemann integral to manage. The Lebesgue integral also works better in many mathematical studies. For example, under simple conditions, it is easier to work with limits when using Lebesgue integration compared to the Riemann integral.

The term Lebesgue integration can refer to the general theory of integrating a function using a measure, as introduced by Lebesgue, or to the specific case of integrating a function on part of the real line using 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 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 with certain 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.

In simple terms, the Lebesgue integral looks at how values of the function are grouped, which helps with more complex functions.

Definition

The Lebesgue integral is a way to find the area under a curve. It works for more complicated functions than the usual method, called the Riemann integral. A French mathematician named Henri Lebesgue created this method.

We can build the Lebesgue integral using simple functions. These are functions that only have a few different values. By breaking a complicated function into simpler pieces, we can find the integral step by step. This helps us understand integrals better.

Main article: Layer cake representation

Example

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

However, the Lebesgue method can integrate this function. It looks at how much space the rational numbers take up. Since there are only countably many rational numbers, their total space is zero. So, the Lebesgue integral of this function from 0 to 1 equals zero.

Domain of integration

In Lebesgue integration, we find the size of a shape without worrying about direction. In basic calculus, integrating from point b to a gives the opposite result of integrating from a to b. Lebesgue integration uses something called a measure to find the size of parts of space. This way, we can calculate over any chosen area, not just straight lines. This idea helps us work with shapes in higher dimensions and connects to a theory called homological integration, made 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 is useful but has some problems with certain math tasks. One issue is when we try to change the order of steps in calculating and integrating functions. This can be tricky and doesn't always work well with the Riemann method.

The Riemann integral also has trouble with some special functions and shapes. It works best with simple shapes and functions that behave nicely, but it can fail with more complex functions or unusual areas. While there are ways to extend the Riemann integral, many functions are still too hard for it. The Lebesgue integral can handle more of these functions. Also, the Riemann integral is closely tied to the natural order of numbers, which limits its use in more complex math 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. This means they differ only on a very small set of points. The Lebesgue integral has useful properties.

One property is linearity. This means the integral of a sum of functions equals the sum of their integrals. Another property is monotonicity. This means 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 think about the Lebesgue integral without using complex ideas. One method is called the Daniell integral.

Another way uses ideas from functional analysis. We start with simple functions and build up to more complicated ones. This helps us define the Lebesgue integral clearly.

Limitations of Lebesgue integral

The Lebesgue integral helps us add up areas under curves, even for very complex shapes. But not every function can be handled this way.

For example, the sinc function looks like a wave that gets smaller and smaller as it moves away from the center. The Lebesgue method cannot fully integrate this function 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.