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 , built on these concepts, generalizes the . 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
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A function f:X→R is measurable if for every open set G⊂R, the preimage f−1(G) is a 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⋅g, and f/g (where g=0) are also measurable
This property allows for the construction of new measurable functions from existing ones
Examples:
If f(x)=x2 and g(x)=sin(x) are measurable, then f+g, f−g, f⋅g, and f/g (where g=0) are also measurable
Composition and Limit Properties
The composition of a measurable function with a continuous function is measurable
If f:X→R is measurable and g:R→R is continuous, then g∘f:X→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)=x2 is measurable and g(x)=x is continuous, then g∘f(x)=∣x∣ is measurable
If {fn} 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 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 α,β∈R, then αf+βg is a measurable function
If f and g are measurable functions, then f⋅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→[0,∞], the Lebesgue integral is defined as the supremum of the integrals of simple functions that are less than or equal to f
A 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 ∫fdμ, where μ is the measure on X
Examples:
The Lebesgue integral of the indicator function 1A of a measurable set A is equal to the measure of A: ∫1Adμ=μ(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 ∫fdμ=∫f+dμ−∫f−dμ, 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 [0,2π], then f+(x)=max{sin(x),0} and f−(x)=max{−sin(x),0}, and ∫02πsin(x)dx=∫02πf+(x)dx−∫02πf−(x)dx=0
If f(x)=x on [−1,1], then f+(x)=max{x,0} and f−(x)=max{−x,0}, and ∫−11xdx=∫−11f+(x)dx−∫−11f−(x)dx=0
Convergence Theorems for Lebesgue Integrals
Monotone Convergence Theorem
The states that if {fn} is a sequence of non-negative measurable functions that is monotonically increasing (i.e., fn≤fn+1 for all n) and converges pointwise to a function f, then lim∫fndμ=∫fdμ
This theorem allows for the interchange of limits and integrals for monotonically increasing sequences of non-negative measurable functions
Examples:
If fn(x)=1−e−nx for x≥0, then {fn} is a monotonically increasing sequence of non-negative measurable functions converging pointwise to f(x)=1 for x≥0, and lim∫0∞fn(x)dx=∫0∞f(x)dx=∞
If fn(x)=min{x2,n} on [0,1], then {fn} is a monotonically increasing sequence of non-negative measurable functions converging pointwise to f(x)=x2 on [0,1], and lim∫01fn(x)dx=∫01f(x)dx=31
Dominated Convergence Theorem
The states that if {fn} is a sequence of measurable functions that converges pointwise to a function f and there exists a non-negative integrable function g such that ∣fn∣≤g for all n, then lim∫fndμ=∫fdμ
This theorem allows for the interchange of limits and integrals for sequences of measurable functions that are dominated by an integrable function
Examples:
If fn(x)=1+nx2x on R, then {fn} converges pointwise to f(x)=0 and is dominated by g(x)=x1 on [1,∞), and lim∫1∞fn(x)dx=∫1∞f(x)dx=0
If fn(x)=nsin(nx) on [0,π], then {fn} converges pointwise to f(x)=0 and is dominated by g(x)=n1 on [0,π], and lim∫0πfn(x)dx=∫0πf(x)dx=0
Fatou's Lemma
is another important result related to the convergence of the Lebesgue integral. It states that if {fn} is a sequence of non-negative measurable functions, then ∫liminffndμ≤liminf∫fndμ
This lemma provides a lower bound for the limit inferior of the integrals of a sequence of non-negative measurable functions
Examples:
If fn(x)=n1(0,n1)(x) on [0,1], then liminffn(x)=0 for all x∈[0,1], and ∫01liminffn(x)dx=0≤liminf∫01fn(x)dx=1
If fn(x)=e−nx on [0,∞), then liminffn(x)=0 for all x∈[0,∞), and ∫0∞liminffn(x)dx=0≤liminf∫0∞fn(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 (Ω,F,P), the expectation of X is defined as E[X]=∫XdP, 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 E[X]=∫−∞∞xf(x)dx
If X is a discrete random variable with probability mass function p, then E[X]=∑xxp(x)
Functional Analysis and Fourier Analysis
The Lebesgue integral is used to define the Lp spaces, which are important function spaces in functional analysis and Fourier analysis
For 1≤p<∞, the Lp space is defined as the set of measurable functions f such that ∫∣f∣pdμ<∞, with the norm ∥f∥p=(∫∣f∣pdμ)1/p
The L∞ space is defined as the set of measurable functions f such that there exists a constant C with ∣f∣≤C almost everywhere, with the norm ∥f∥∞=inf{C:∣f∣≤C almost everywhere}
In Fourier analysis, the Lebesgue integral is used to define the Fourier transform of functions in Lp spaces
For f∈L1(R), the Fourier transform of f is defined as f^(ξ)=∫f(x)e−2πixξdx, where the integral is a Lebesgue integral
Examples:
The space of square-integrable functions L2([0,1]) is a Hilbert space with inner product ⟨f,g⟩=∫01f(x)g(x)dx
The Fourier transform of the Gaussian function f(x)=e−πx2 is given by f^(ξ)=e−πξ2
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 −Δu=f on a domain Ω with boundary conditions u=0 on ∂Ω is given by ∫Ω∇u⋅∇vdx=∫Ωfvdx for all test functions v∈H01(Ω)
The existence and uniqueness of weak solutions to the heat equation ∂t∂u−Δu=f on a domain Ω with initial and boundary conditions can be proven using the Lebesgue integral and the Lax-Milgram theorem
Key Terms to Review (18)
Almost Everywhere Convergence: Almost everywhere convergence refers to the behavior of a sequence of functions that converges to a limit function at all points in a measure space, except for a set of points with measure zero. This concept is crucial in understanding the properties of measurable functions and how they interact with integration. It highlights the idea that the convergence can be disregarded on negligible sets, allowing for meaningful analysis in spaces where traditional convergence may not hold.
Anders W. Thorbjørnsen: Anders W. Thorbjørnsen is a mathematician known for his contributions to geometric measure theory, particularly in the area of measurable functions and integration. His work often focuses on the intricate relationships between geometric properties and analytical techniques, providing crucial insights into how measurable functions can be understood and utilized within various mathematical contexts.
Borel measurable function: A Borel measurable function is a function defined on a measurable space that takes values in a topological space and is measurable with respect to the Borel $oldsymbol{\sigma}$-algebra. These functions map Borel sets to measurable sets, preserving the structure necessary for integration and analysis. Understanding Borel measurable functions is essential for exploring properties of measurable spaces and integrating over them, as they ensure that we can work effectively with limits, continuity, and convergence in mathematical analysis.
Dominated Convergence Theorem: The Dominated Convergence Theorem is a fundamental result in measure theory that provides conditions under which the limit of an integral can be interchanged with the limit of a sequence of functions. This theorem is crucial for working with measurable functions and integration, as it ensures that if a sequence of functions converges pointwise to a limit and is dominated by an integrable function, then the integral of the limit equals the limit of the integrals of those functions.
Fatou's Lemma: Fatou's Lemma is a fundamental result in measure theory that provides a relationship between the limit of integrals of non-negative measurable functions and the integral of their pointwise limit. It states that for a sequence of non-negative measurable functions, the integral of the limit inferior is less than or equal to the limit inferior of the integrals. This lemma is essential in understanding convergence in integration, especially in contexts involving measurable functions.
Fubini's Theorem: Fubini's Theorem states that for a measurable function defined on a product measure space, the double integral of that function can be computed as an iterated integral. This theorem is crucial for switching the order of integration in multiple integrals, allowing us to integrate first with respect to one variable and then with respect to another. Its application simplifies many problems in measure theory and helps establish connections between measurable functions and integration, as well as the area and coarea formulas.
Henri Léon Lebesgue: Henri Léon Lebesgue was a French mathematician best known for developing the concept of measure theory and the Lebesgue integral, which revolutionized the way we understand integration and measure in mathematical analysis. His work laid the foundation for modern probability theory and has profound implications in various areas, especially concerning measurable functions, the properties of Hausdorff measure, and isoperimetric inequalities.
Lebesgue Integral: The Lebesgue integral is a method of integration that extends the concept of the integral to a broader class of functions and allows for the integration of functions defined on measurable sets. It is based on the idea of measuring the size of sets and sums, facilitating the handling of limits and convergence in a more flexible way compared to traditional Riemann integration. This integral is crucial for working with measurable functions and provides the foundation for various applications in probability, real analysis, and geometric measure theory.
Lebesgue Measurable Function: A Lebesgue measurable function is a function defined on a measurable space that takes values in the real numbers and is measurable with respect to the Lebesgue measure. This means that the preimage of any Lebesgue measurable set under the function is also a Lebesgue measurable set. These functions play a vital role in the integration theory, as they ensure that we can meaningfully integrate them over measurable sets.
Measurable set: A measurable set is a subset of a given space that can be assigned a meaningful size or measure, typically in the context of Lebesgue measure. These sets allow for the development of integration and analysis on functions defined over them, ensuring that important properties like countable unions and intersections hold true within the measure theory framework.
Measurable Space: A measurable space is a mathematical structure that consists of a set along with a sigma-algebra on that set, which defines a collection of subsets considered to be measurable. This framework is crucial for establishing a foundation for measures, which quantify the size or probability of these subsets. Measurable spaces enable the study of functions and integration by allowing us to explore properties such as convergence, continuity, and the integration of measurable functions over defined sets.
Measure Space: A measure space is a mathematical structure that provides a way to assign a size or measure to subsets of a given set, where this size must satisfy certain properties. It is composed of a set, a sigma-algebra of subsets of that set, and a measure function that assigns a non-negative real number or infinity to each subset in the sigma-algebra. This concept is crucial for understanding measurable functions and integration, as well as for exploring geometric measure theory in more complex structures like metric measure spaces.
Monotone Convergence Theorem: The Monotone Convergence Theorem states that if you have a sequence of measurable functions that are non-decreasing and converge pointwise to a limit function, then the integral of these functions converges to the integral of the limit function. This theorem is essential because it provides a powerful way to exchange limits and integrals, making it easier to analyze the behavior of integrals of functions over time. It emphasizes the importance of the conditions under which these mathematical operations can be interchanged.
Null Set: A null set, also known as an empty set, is a set that contains no elements. In measure theory, it is crucial because it plays a key role in defining measures and understanding properties of measurable spaces, specifically when measuring sets that may not contribute to the overall measure, thus having a measure of zero. The concept of a null set is vital in distinguishing between measurable and non-measurable sets, as well as in the analysis of functions and their properties like continuity and integrability.
Pointwise Convergence: Pointwise convergence refers to a sequence of functions converging at each point in their domain individually. In this context, if a sequence of functions converges pointwise to a limit function, it means that for every point in the domain, the values of the sequence of functions approach the value of the limit function as the sequence progresses. This concept is essential in understanding the behavior of sequences of measurable functions, their integrals, and how they relate to various properties like regularity and Lipschitz conditions.
Riemann Integral: The Riemann integral is a method of assigning a number to the area under a curve defined by a function over a specified interval. This integral is calculated using the concept of partitioning the interval into subintervals, determining the function's value at specific points, and summing the resulting areas of rectangles formed, which approaches the actual area as the partition gets finer. Understanding the Riemann integral is crucial in analyzing measurable functions and their properties in integration.
Sigma-algebra: A sigma-algebra is a collection of sets that is closed under the operations of countable unions, countable intersections, and complements. It provides a structured way to define measurable spaces, allowing for the rigorous development of measures and integration, which are foundational in probability and analysis.
Simple Function: A simple function is a type of measurable function that takes on only a finite number of values, making it easier to analyze in the context of integration and measure theory. Simple functions are typically expressed as finite sums of characteristic functions multiplied by constants, allowing for straightforward integration over measurable sets. Their structure serves as a foundational tool for approximating more complex functions, particularly in the development of the Lebesgue integral.