Measurable functions and integration form the backbone of measure theory. They extend the concept of measurability from sets to functions, allowing us to work with a broader class of mathematical objects. This extension is crucial for developing a more powerful and flexible integration theory.

The Lebesgue integral, built on these concepts, generalizes the Riemann integral. It can handle a wider range of functions, including those with discontinuities or unbounded values. This makes it an essential tool in advanced mathematics, particularly in analysis and probability theory.

Measurable Functions and Properties

Definition and Preimage Property

• A function $f: X \to \mathbb{R}$ is measurable if for every open set $G \subset \mathbb{R}$, the preimage $f^{-1}(G)$ is a measurable set in $X$
• This property allows for the extension of measurability from sets to functions
• Examples of measurable functions include continuous functions, step functions, and piecewise continuous functions

Preservation of Measurability under Arithmetic Operations

• The sum, difference, product, and quotient of two measurable functions are also measurable
  • If $f$ and $g$ are measurable functions, then $f + g$, $f - g$, $f \cdot g$, and $f/g$ (where $g \neq 0$) are also measurable
• This property allows for the construction of new measurable functions from existing ones
• Examples:
  • If $f(x) = x^2$ and $g(x) = \sin(x)$ are measurable, then $f + g$, $f - g$, $f \cdot g$, and $f/g$ (where $g \neq 0$) are also measurable

Composition and Limit Properties

• The composition of a measurable function with a continuous function is measurable
  • If $f: X \to \mathbb{R}$ is measurable and $g: \mathbb{R} \to \mathbb{R}$ is continuous, then $g \circ f: X \to \mathbb{R}$ is measurable
• The limit of a sequence of measurable functions, when it exists pointwise, is also measurable
• These properties allow for the creation of new measurable functions through composition and limits
• Examples:
  • If $f(x) = x^2$ is measurable and $g(x) = \sqrt{x}$ is continuous, then $g \circ f(x) = |x|$ is measurable
  • If ${f_n}$ is a sequence of measurable functions converging pointwise to $f$, then $f$ is also measurable

Vector Space and Algebra Structure

• The set of measurable functions forms a vector space over $\mathbb{R}$ under pointwise addition and scalar multiplication
• The set of measurable functions also forms an algebra under pointwise addition and multiplication
• These algebraic structures provide a framework for studying measurable functions and their properties
• Examples:
  • If $f$ and $g$ are measurable functions and $\alpha, \beta \in \mathbb{R}$, then $\alpha f + \beta g$ is a measurable function
  • If $f$ and $g$ are measurable functions, then $f \cdot g$ is a measurable function

Lebesgue Integral for Non-negative Functions

Definition and Motivation

• The Lebesgue integral is a generalization of the Riemann integral that allows for the integration of a broader class of functions, including unbounded and discontinuous functions
• For a non-negative measurable function $f: X \to [0, \infty]$, the Lebesgue integral is defined as the supremum of the integrals of simple functions that are less than or equal to $f$
  • A simple function is a measurable function that takes on a finite number of values
• The Lebesgue integral of a non-negative measurable function $f$ is denoted as $\int f d\mu$, where $\mu$ is the measure on $X$
• Examples:
  • The Lebesgue integral of the indicator function $\mathbf{1}_A$ of a measurable set $A$ is equal to the measure of $A$: $\int \mathbf{1}_A d\mu = \mu(A)$
  • The Lebesgue integral of a non-negative continuous function $f$ on a compact interval $[a, b]$ is equal to the Riemann integral of $f$ on $[a, b]$

Extension to General Measurable Functions

• The Lebesgue integral can be extended to measurable functions that take on both positive and negative values by splitting the function into its positive and negative parts
• For a measurable function $f$, define $f^+(x) = \max{f(x), 0}$ and $f^-(x) = \max{-f(x), 0}$. Then, $f = f^+ - f^-$, and the Lebesgue integral of $f$ is defined as $\int f d\mu = \int f^+ d\mu - \int f^- d\mu$, provided that at least one of the integrals on the right-hand side is finite
• This extension allows for the integration of a wide range of measurable functions, including those with both positive and negative values
• Examples:
  • If $f(x) = \sin(x)$ on $[0, 2\pi]$, then $f^+(x) = \max{\sin(x), 0}$ and $f^-(x) = \max{-\sin(x), 0}$, and $\int_0^{2\pi} \sin(x) dx = \int_0^{2\pi} f^+(x) dx - \int_0^{2\pi} f^-(x) dx = 0$
  • If $f(x) = x$ on $[-1, 1]$, then $f^+(x) = \max{x, 0}$ and $f^-(x) = \max{-x, 0}$, and $\int_{-1}^1 x dx = \int_{-1}^1 f^+(x) dx - \int_{-1}^1 f^-(x) dx = 0$

Convergence Theorems for Lebesgue Integrals

Monotone Convergence Theorem

• The Monotone Convergence Theorem states that if ${f_n}$ is a sequence of non-negative measurable functions that is monotonically increasing (i.e., $f_n \leq f_{n+1}$ for all $n$) and converges pointwise to a function $f$, then $\lim \int f_n d\mu = \int f d\mu$
• This theorem allows for the interchange of limits and integrals for monotonically increasing sequences of non-negative measurable functions
• Examples:
  • If $f_n(x) = 1 - e^{-nx}$ for $x \geq 0$, then ${f_n}$ is a monotonically increasing sequence of non-negative measurable functions converging pointwise to $f(x) = 1$ for $x \geq 0$, and $\lim \int_0^\infty f_n(x) dx = \int_0^\infty f(x) dx = \infty$
  • If $f_n(x) = \min{x^2, n}$ on $[0, 1]$, then ${f_n}$ is a monotonically increasing sequence of non-negative measurable functions converging pointwise to $f(x) = x^2$ on $[0, 1]$, and $\lim \int_0^1 f_n(x) dx = \int_0^1 f(x) dx = \frac{1}{3}$

Dominated Convergence Theorem

• The Dominated Convergence Theorem states that if ${f_n}$ is a sequence of measurable functions that converges pointwise to a function $f$ and there exists a non-negative integrable function $g$ such that $|f_n| \leq g$ for all $n$, then $\lim \int f_n d\mu = \int f d\mu$
• This theorem allows for the interchange of limits and integrals for sequences of measurable functions that are dominated by an integrable function
• Examples:
  • If $f_n(x) = \frac{x}{1 + nx^2}$ on $\mathbb{R}$, then ${f_n}$ converges pointwise to $f(x) = 0$ and is dominated by $g(x) = \frac{1}{x}$ on $[1, \infty)$, and $\lim \int_1^\infty f_n(x) dx = \int_1^\infty f(x) dx = 0$
  • If $f_n(x) = \frac{\sin(nx)}{n}$ on $[0, \pi]$, then ${f_n}$ converges pointwise to $f(x) = 0$ and is dominated by $g(x) = \frac{1}{n}$ on $[0, \pi]$, and $\lim \int_0^\pi f_n(x) dx = \int_0^\pi f(x) dx = 0$

Fatou's Lemma

• Fatou's Lemma is another important result related to the convergence of the Lebesgue integral. It states that if ${f_n}$ is a sequence of non-negative measurable functions, then $\int \liminf f_n d\mu \leq \liminf \int f_n d\mu$
• This lemma provides a lower bound for the limit inferior of the integrals of a sequence of non-negative measurable functions
• Examples:
  • If $f_n(x) = n \mathbf{1}_{(0, \frac{1}{n})}(x)$ on $[0, 1]$, then $\liminf f_n(x) = 0$ for all $x \in [0, 1]$, and $\int_0^1 \liminf f_n(x) dx = 0 \leq \liminf \int_0^1 f_n(x) dx = 1$
  • If $f_n(x) = e^{-nx}$ on $[0, \infty)$, then $\liminf f_n(x) = 0$ for all $x \in [0, \infty)$, and $\int_0^\infty \liminf f_n(x) dx = 0 \leq \liminf \int_0^\infty f_n(x) dx = 0$

Importance in Proving Properties and Interchange of Limits

• These convergence theorems are crucial for proving the properties of the Lebesgue integral and for justifying the interchange of limits and integrals in various situations
• They provide a solid foundation for the study of the Lebesgue integral and its applications in different areas of mathematics
• Examples:
  • The Monotone Convergence Theorem can be used to prove the linearity and monotonicity of the Lebesgue integral
  • The Dominated Convergence Theorem can be used to justify the interchange of limits and integrals in the definition of the Fourier transform and in the study of weak solutions of partial differential equations

Applications of Lebesgue Integration

Probability Theory

• In probability theory, the Lebesgue integral is used to define the expectation of random variables
  • For a random variable $X$ on a probability space $(\Omega, \mathcal{F}, P)$, the expectation of $X$ is defined as $\mathbb{E}[X] = \int X dP$, provided that the integral exists
• The Lebesgue integral allows for the definition of expectation for a wide range of random variables, including those with unbounded or discontinuous distributions
• Examples:
  • If $X$ is a continuous random variable with probability density function $f$, then $\mathbb{E}[X] = \int_{-\infty}^\infty x f(x) dx$
  • If $X$ is a discrete random variable with probability mass function $p$, then $\mathbb{E}[X] = \sum_{x} x p(x)$

Functional Analysis and Fourier Analysis

• The Lebesgue integral is used to define the $L^p$ spaces, which are important function spaces in functional analysis and Fourier analysis
  • For $1 \leq p < \infty$, the $L^p$ space is defined as the set of measurable functions $f$ such that $\int |f|^p d\mu < \infty$, with the norm $|f|_p = (\int |f|^p d\mu)^{1/p}$
  • The $L^\infty$ space is defined as the set of measurable functions $f$ such that there exists a constant $C$ with $|f| \leq C$ almost everywhere, with the norm $|f|_\infty = \inf{C : |f| \leq C \text{ almost everywhere}}$
• In Fourier analysis, the Lebesgue integral is used to define the Fourier transform of functions in $L^p$ spaces
  • For $f \in L^1(\mathbb{R})$, the Fourier transform of $f$ is defined as $\hat{f}(\xi) = \int f(x)e^{-2\pi ix\xi} dx$, where the integral is a Lebesgue integral
• Examples:
  • The space of square-integrable functions $L^2([0, 1])$ is a Hilbert space with inner product $\langle f, g \rangle = \int_0^1 f(x) \overline{g(x)} dx$
  • The Fourier transform of the Gaussian function $f(x) = e^{-\pi x^2}$ is given by $\hat{f}(\xi) = e^{-\pi \xi^2}$

Partial Differential Equations

• The Lebesgue integral plays a role in the study of partial differential equations, where it is used to define weak solutions and to prove existence and uniqueness results
• Weak solutions are defined using the Lebesgue integral and allow for the study of solutions that may not be differentiable in the classical sense
• Examples:
  • The weak formulation of the Poisson equation $-\Delta u = f$ on a domain $\Omega$ with boundary conditions $u = 0$ on $\partial \Omega$ is given by $\int_\Omega \nabla u \cdot \nabla v dx = \int_\Omega fv dx$ for all test functions $v \in H_0^1(\Omega)$
  • The existence and uniqueness of weak solutions to the heat equation $\frac{\partial u}{\partial t} - \Delta u = f$ on a domain $\Omega$ with initial and boundary conditions can be proven using the Lebesgue integral and the Lax-Milgram theorem