Lebesgue Integration

Lebesgue integration is a fundamental concept in real analysis and has profound applications in a variety of mathematical fields, including probability theory, functional analysis, and beyond. Compared to Riemann integration, the Lebesgue integral provides a more powerful and versatile framework for integrating functions, especially where the Riemann integral falls short. This lesson will walk you through the essential ideas and construction of the Lebesgue integral using simple language, examples, and illustrative visualizations.

Introduction to integration concepts

At its core, integration is about accumulating quantities over a certain domain. For example, if you think of a curve on a graph, integration calculates the area under that curve over a given interval. The Riemann integral, which you may be familiar with, involves dividing the domain into intervals and adding rectangles to estimate the area under the curve.

Challenge with Riemann integration

While the Riemann integral works well for many functions, it struggles with some functions, particularly those that have a lot of discontinuities or that are defined in an infinite number of points but do not behave well throughout time.

Setting the stage: Measurable actions

To understand Lebesgue integration, we first need to introduce some key mathematical ideas: measure and measurable functions.

Measurement theory basics

In a broad sense, a measure is a systematic way of assigning numbers representing size to subsets of a given set. The most familiar measure is the length of intervals on the real line, or the "Lebesgue measure". A set is said to be measurable if its length can be well defined in this sense.

Introduction to the Lebesgue integral

The Lebesgue integral focuses on measuring vertical "slices" of the function rather than horizontal divisions of the domain, as in the Riemann case. This process is broken down into the following key concepts:

• Simple functions: These are functions that take a finite number of values, often using indicator functions on measurable sets. They are the integral building blocks of the Lebesgue integral.
• Measurable function: If the pre-image of every interval is a measurable set, then the function is measurable. This is an important pre-condition for Lebesgue integration.
• Lebesgue integration of simple functions: The integral of a simple function is computed on every measurable set as the sum of the product of the value of the function and the measure of the set.

With these preliminaries, we can formally define the Lebesgue integral for more complicated functions by understanding them as limits of sequences of simpler functions.

Simple function example

Suppose ( f(x) ) is a simple function defined on the interval [0, 1]:
f(x) = 
  begin{cases}
    3, & text{if $x in [0, 0.5)$} 
    7, & text{if $x in [0.5, 1]$}
  end{cases}

The integral of this simple function on [0, 1] using Lebesgue integration is:

∫ from 0 to 1 ( f(x) , dx ) = 3 × m([0,0.5)) + 7 × m([0.5,1]) = 3×0.5 + 7×0.5 = 5

Construction of the Lebesgue integral

The main idea of the Lebesgue integral is to decompose the image of a function into parts and measure how much an input contributes to those output parts. Let's break down this process:

Integration of non-negative measurable functions

For a non-negative measurable function (f) on a measurable set (E), we can define Lebesgue integration as follows:

• Represent ( f ) as the limit of an increasing sequence of simple functions ( f_n ) such that ( f_n to f ) is pointwise.
• Integrate each simple function ( f_n ) over ( E ): ( int_E f_n(x) , dx ).
• The Lebesgue integral is the limit of these integrals: ∫_E f(x) , dx = lim_{n→∞} ∫_E f_n(x) , dx

Examples of Lebesgue integration

Example 1: Step function

The step function jumps between values and can be viewed as a simple function. Consider:

f(x) = 
  begin{cases}
    2, & text{if $x in [0, 1]$} 
    5, & text{if $x in (1, 3]$}
  end{cases}

To find the Lebesgue integral over ([0,3]), calculate:

∫ from 0 to 3 ( f(x) , dx ) = 2×m([0,1]) + 5×m((1,3]) = 2×1 + 5×2 = 12

Example 2: Indicator function

The indicator function (chi_A(x)) of a set A is defined as:

 chi_A(x) = 
  begin{cases}
    1, & text{if $x in A$} \
    0, & text{otherwise}
  end{cases}

Let's assume A is measurable and we want to integrate over ([0,2]):

∫ from 0 to 2 ( chi_A(x) , dx = m(A cap [0,2]) )

Extension to general functions

For any real-valued function, we can express it as the difference of two non-negative functions:

f(x) = f^+(x) - f^-(x)

Where:

f^+(x) = max(f(x), 0)
f^-(x) = max(-f(x), 0)

Then we can define the Lebesgue integral of ( f ) as follows:

∫_E f(x) , dx = ∫_E f^+(x) , dx - ∫_E f^-(x) , dx

Comparison: Lebesgue vs. Riemann integrals

To appreciate the versatility of the Lebesgue integral, let's compare it to the Riemann integral:

• Scope of functions: Lebesgue integration can handle a wide class of functions, including functions with an infinite number of discontinuities.
• Convergence: Lebesgue's maximal convergence theorem provides tools for interchanging limits and integrals in some circumstances, which are more restrictive than the Riemann sense.
• Measure theory: Lebesgue integrates based on measurements, making it suitable for probability and statistical applications where measurements determine behavior.

Dominant convergence theorem

One of the notable results that distinguishes Lebesgue integration is the Dominated Convergence Theorem (DCT). It states that if a sequence of measurable functions ( f_n(x) ) converges pointwise to ( f(x) ) then and is dominated by an integrable function ( g(x) ) (i.e., ( |f_n(x)| leq g(x) ) for all ( n )), then:

lim_{n→∞} ∫_E f_n(x) , dx = ∫_E lim_{n→∞} f_n(x) , dx

This flexibility is extremely useful in various areas of analysis where we deal with limits and we want to take the limits under the integral sign, without stepping outside the realm of integrable functions.

Visual example

Conclusion

Lebesgue integration expands our mathematical toolkit, providing a powerful way to address tasks that cannot be addressed by Riemann integration. Through its reliance on measure theory and its focus on function value distributions rather than domain partitions, Lebesgue integration expands our mathematical toolkit, providing a powerful way to address tasks that cannot be addressed by Riemann integration. In turn, it opens the doors to more sophisticated analysis, particularly in spaces such as probability and measure-based calculus. This approach enriches our understanding, providing a coherent framework that can be used to describe non-deterministic, discontinuous, or otherwise complex functions. Makes it easier to work with.

Becoming familiar with Lebesgue integration is a significant leap forward in understanding analysis in a deeper way, providing weapons to tackle real-world applications where nature does not always work according to simple rules.

Comments