The Ultimate Resource for Lebesgue Integration on Euclidean Space: Frank Jones' PDF
Lebesgue Integration on Euclidean Space: A Review of Jones' Book
Lebesgue integration is a powerful mathematical tool that allows us to integrate functions that are not necessarily continuous or even bounded on Euclidean space. It also provides a framework for studying various topics in analysis, such as measure theory, differentiation, functions of bounded variation, Lp spaces, and Fourier analysis. In this article, we will review the book Lebesgue Integration on Euclidean Space by Frank Jones, which is a comprehensive and accessible introduction to this subject. We will summarize the main concepts, results, and proofs presented in each chapter, as well as provide some examples and applications. We will also highlight some of the strengths and weaknesses of the book, as well as some suggestions for further reading.
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Introduction
In this chapter, Jones motivates the need for Lebesgue integration by showing some limitations of Riemann integration. He gives an example of a function that is Riemann integrable but whose integral depends on the choice of partition. He also gives an example of a function that is not Riemann integrable but whose integral can be defined using Lebesgue's approach. He then explains the main idea behind Lebesgue integration, which is to measure the area under the graph of a function by counting how many horizontal slices of a given height fit under the graph. He also introduces some basic notation and terminology that will be used throughout the book.
What is Lebesgue integration?
Lebesgue integration is a generalization of Riemann integration that allows us to integrate functions that are not necessarily continuous or even bounded on a given domain. The key idea is to define the integral of a function f as the sum of the areas of horizontal rectangles whose heights are determined by the values of f and whose widths are determined by the sizes of the subsets of the domain where f takes those values. This way, we can integrate functions that have discontinuities, jumps, or spikes, as long as they are measurable, which means that we can assign a meaningful size to the subsets of the domain where they take certain values.
Why is it important for Euclidean space?
Lebesgue integration is important for Euclidean space because it allows us to study many functions and phenomena that arise in geometry, physics, probability, and other fields. For example, Lebesgue integration enables us to define and compute the length, area, and volume of various shapes and solids, even if they have curved or fractal boundaries. It also enables us to define and compute the average value, variance, and expectation of random variables and processes, even if they have discontinuous or infinite outcomes. It also enables us to analyze the convergence, continuity, and differentiability of functions and series, even if they have irregular or oscillatory behavior.
What are the main topics covered in Jones' book?
Jones' book covers six main topics related to Lebesgue integration on Euclidean space. They are:
Measure theory: the study of how to assign sizes to subsets of Euclidean space, such as length, area, and volume.
Lebesgue measure and integration: the construction and properties of the standard measure and integral on Euclidean space.
Differentiation and integration: the relation between differentiation and integration, and the Lebesgue differentiation theorem.
Functions of bounded variation and rectifiable curves: the characterization and properties of functions that have finite total variation, and the relation between such functions and curves that have finite length.
Lp spaces: the definition and properties of spaces of functions that have finite p-norms, which measure how large or small a function is in terms of its absolute values.
Fourier series and Fourier transform: the definition and properties of series and transform that decompose a function into its frequency components, which reveal how a function behaves in terms of its periodicity or symmetry.
Chapter 1: Measure Theory
In this chapter, Jones introduces the concept of measure, which is a way of assigning sizes to subsets of Euclidean space. He explains how to define a measure on a given set using an outer measure, which is a function that gives an upper bound for the size of any subset. He then shows how to check if a subset is measurable, which means that its outer measure agrees with its inner measure, which is a function that gives a lower bound for the size of any subset. He also shows how to construct an outer measure using a countable collection of sets that cover the whole space, such as intervals or cubes. He then proves some basic properties of measures, such as monotonicity, subadditivity, countable additivity, and continuity.
What is a measure and how is it defined?
A measure is a function that assigns a non-negative number to each subset of a given set, such as Euclidean space. The number represents the size of the subset in some sense, such as length, area, or volume. A measure must satisfy two axioms:
Non-negativity: The measure of any subset is non-negative.
Countable additivity: The measure of a countable union of disjoint subsets is equal to the sum of their measures.
A measure can be defined on a given set using an outer measure, which is a function that assigns a non-negative number to any subset of the set. The number represents an upper bound for the size of the subset. An outer measure must satisfy two axioms:
Non-negativity: The outer measure of any subset is non-negative.
Monotonicity: The outer measure of a subset is less than or equal to the outer measure of any superset.
An outer measure can be used to define a measure on a subset if the subset is measurable, which means that its outer measure agrees with its inner measure. The inner measure is a function that assigns a non-negative number to any subset of the set. The number represents a lower bound for the size of the subset. An inner measure can be defined using an outer measure as follows:
The inner measure of a subset A is equal to the complement of the outer measure of the complement of A. That is,
$$ m_*(A) = m(X) - m^*(X \setminus A) $$ where X is the whole set, m is the outer measure, and m* is the inner measure.
What are some examples of measures on Euclidean space?
One of the most important and common measures on Euclidean space is the Lebesgue measure, which is the standard way of measuring length, area, and volume of subsets of Euclidean space. The Lebesgue measure is defined using an outer measure that is based on covering subsets with countable collections of intervals or cubes. The Lebesgue measure has many desirable properties, such as translation invariance, rotation invariance, and scale invariance. The Lebesgue measure will be discussed in more detail in Chapter 2.
Another example of a measure on Euclidean space is the counting measure, which is a simple way of measuring the size of a subset by counting how many elements it contains. The counting measure is defined using an outer measure that assigns 1 to any singleton set and infinity to any other set. The counting measure is useful for studying discrete sets and functions, such as sequences and series.
A third example of a measure on Euclidean space is the Hausdorff measure, which is a generalization of the Lebesgue measure that can measure subsets of any dimension, even if they are not integer-valued. The Hausdorff measure is defined using an outer measure that is based on covering subsets with countable collections of balls or spheres. The Hausdorff measure can be used to measure the size of fractals and other irregular shapes.
What are some properties of measures and how are they proved?
Some of the basic properties of measures are:
Subadditivity: The measure of a union of subsets is less than or equal to the sum of their measures.
Continuity from below: If a sequence of subsets increases to a limit subset, then the measure of the limit subset is equal to the limit of the measures of the subsets.
Continuity from above: If a sequence of subsets decreases to a limit subset, and the first subset has finite measure, then the measure of the limit subset is equal to the limit of the measures of the subsets.
These properties can be proved using the definition and axioms of measures and outer measures. For example, to prove subadditivity, we can use the fact that an outer measure is monotonic and countably subadditive. That is,
$$ m(A \cup B) \leq m^*(A \cup B) \leq m^*(A) + m^*(B) = m(A) + m(B) $$ where A and B are measurable subsets and m and m* are the measure and outer measure respectively.
Chapter 2: Lebesgue Measure and Integration
In this chapter, Jones constructs and studies the Lebesgue measure and integral on Euclidean space. He shows how to define the Lebesgue measure using an outer measure that is based on covering subsets with countable collections of intervals or cubes. He then shows how to define the Lebesgue integral using a simple function, which is a function that takes only finitely many values. He then shows how to extend the Lebesgue integral to more general functions using limits and approximations. He also proves some basic properties of the Lebesgue measure and integral, such as translation invariance, rotation invariance, scale invariance, linearity, monotonicity, and dominated convergence.
What is Lebesgue measure and how is it constructed?
Lebesgue measure is the standard way of measuring length, area, and volume of subsets of Euclidean space. It assigns a non-negative number to each measurable subset that represents its size in some sense. For example, the Lebesgue measure of an interval is equal to its length, the Lebesgue measure of a rectangle is equal to its area, and the Lebesgue measure of a cube is equal to its volume.
of intervals or cubes. An interval is a set of the form [a,b], where a and b are real numbers and a
$$ v([a1,b1] x [a2,b2] x ... x [an,bn]) = (b1 - a1)(b2 - a2)...(bn - an) $$ The outer measure based on intervals or cubes is defined as follows:
The outer measure of a subset A of Euclidean space is equal to the infimum of the sum of the volumes of the cubes that cover A. That is,
$$ m^*(A) = \inf \left\ \sum_i=1^\infty v(Q_i) : A \subseteq \bigcup_i=1^\infty Q_i \right\ $$ where Q_i are cubes and the infimum is taken over all possible countable coverings of A by cubes.
The outer measure based on intervals or cubes can be used to define the Lebesgue measure on a subset if the subset is measurable, which means that its outer measure agrees with its inner measure. The inner measure can be defined using the outer measure as follows:
The inner measure of a subset A of Euclidean space is equal to the complement of the outer measure of the complement of A. That is,
$$ m_*(A) = m(\mathbbR^n) - m^*(\mathbbR^n \setminus A) $$ where m is the Lebesgue measure and m* is the outer measure based on intervals or cubes.
What is Lebesgue integration and how is it defined?
Lebesgue integration is a generalization of Riemann integration that allows us to integrate functions that are not necessarily continuous or even bounded on a given domain. It assigns a number to each measurable function that represents the area under its graph in some sense. For example, the Lebesgue integral of a constant function is equal to its value times the measure of its domain, and the Lebesgue integral of an indicator function is equal to the measure of the set where it takes the value 1.
Lebesgue integration can be defined using a simple function, which is a function that takes only finitely many values. The Lebesgue integral of a simple function f is defined as follows:
The Lebesgue integral of f over a measurable set E is equal to the sum of the values of f times the measures of the sets where f takes those values. That is,
$$ \int_E f dm = \sum_i=1^k c_i m(E_i) $$ where c_i are the distinct values of f and E_i are the sets where f(x) = c_i for each i.
The Lebesgue integral of a simple function can be extended to more general functions using limits and approximations. For example, if f is a non-negative measurable function, then we can define its Lebesgue integral as follows:
The Lebesgue integral of f over a measurable set E is equal to the supremum of the Lebesgue integrals of simple functions that are less than or equal to f over E. That is,
$$ \int_E f dm = \sup \left\ \int_E s dm : s \leq f, s \text simple \right\ $$ If f is any measurable function, then we can define its Lebesgue integral as follows:
the Lebesgue integrals of the positive and negative parts of f over E. That is,
$$ \int_E f dm = \int_E f^+ dm - \int_E f^- dm $$ where f^+ and f^- are the positive and negative parts of f, defined as
$$ f^+(x) = \max\f(x), 0\ \quad \textand \quad f^-(x) = \max\-f(x), 0\ $$ What are some properties of Lebesgue integrals and how are they proved?
Some of the basic properties of Lebesgue integrals are:
Linearity: The Lebesgue integral of a linear combination of functions is equal to the linear combination of their Lebesgue integrals.
Monotonicity: If a function is less than or equal to another function, then its Lebesgue integral is less than or equal to the Lebesgue integral of the other function.
Dominated convergence: If a sequence of functions converges pointwise to a limit function, and the sequence is bounded by an integrable function, then the Lebesgue integral of the limit function is equal to the limit of the Lebesgue integrals of the sequence.
These properties can be proved using the definition and properties of simple functions and limits. For example, to prove linearity, we can use the fact that a linear combination of simple functions is a simple function, and that the Lebesgue integral of a simple function is linear. That is,
$$ \int_E (af + bg) dm = \int_E s dm = \sum_i=1^k (ac_i + bc_i') m(E_i) = a \sum_i=1^k c_i m(E_i) + b \sum_i=1^k c_i' m(E_i) = a \int_E f dm + b \int_E g dm $$ where f and g are simple functions, s is their linear combination, c_i and c_i' are their distinct values, and E_i are the sets where s(x) = ac_i + bc_i' for each i.
Chapter 3: Differentiation and Integration
In this chapter, Jones explores the relation between differentiation and integration, and proves the Lebesgue differentiation theorem. He shows how to define the derivative of a function using limits of ratios of differences. He then shows how to define the indefinite integral of a function using antiderivatives. He also shows how to define the definite integral of a function using limits of Riemann sums. He then proves the fundamental theorem of calculus, which states that differentiation and integration are inverse operations under certain conditions. He also proves the Lebesgue differentiation theorem, which states that almost every point of a function is a point of differentiation under certain conditions.
What is the relation between differentiation and integration?
Differentiation and integration are two important operations in calculus that measure how a function changes or accumulates over an interval. Differentiation measures the rate of change or slope of a function at a point, while integration measures the total change or area under a function over an interval. Differentiation and integration are inverse operations under certain conditions, which means that applying one operation after another returns the original function.
What is the Lebesgue differentiation theorem and how is it proved?
The Lebesgue differentiation theorem is a generalization of the fundamental theorem of calculus that applies to Lebesgue integrable functions. It states that almost every point of a function is a point of differentiation under certain conditions. That is,
$$ \lim_r \to 0 \frac1m(B(x,r)) \int_B(x,r) f dm = f(x) $$ for almost every x in Euclidean space, where B(x,r) is a ball centered at x with radius r, m is the Lebesgue measure, and f is an integrable function.
iability, and integration. He also introduces and studies the concept of rectifiable curves, which are curves that have finite length. He shows how to define the length of a curve using the supremum of the sums of distances over partitions. He then shows how to relate rectifiable curves to functions of bounded variation using parametrizations, arc length, and tangents. He also proves some basic properties of functions of bounded variation and rectifiable curves, such as additivity, monotonicity, continuity, and differentiation.
What are functions of bounded variation and how are they characterized?
Functions of bounded variation are functions that have finite total variation, which measures how much a function oscillates or changes over an interval. The total variation of a function f over an interval [a,b] is defined as follows:
The total variation of f over [a,b] is equal to the supremum of the sums of absolute differences of f over all possible partitions of [a,b]. That is,
$$ V_a^b(f) = \sup \left\f(x_i) - f(x_i-1) $$ where the supremum is taken over all possible choices of n and x_i for each i.
Functions of bounded variation can be characterized using several equivalent conditions, such as:
Absolute continuity: A function f is absolutely continuous on [a,b] if for every positive number e there exists a positive number d such that for any finite collection of disjoint subintervals of [a,b], if the sum of the lengths of the subintervals is less than d, then the sum of the absolute differences of f over the subintervals is less than e.
Differentiability: A function f is differentiable almost everywhere on [a,b] and its derivative f' is integrable on [a,b].
Integration: A function f can be written as the difference of two increasing functions on [a,b], or equivalently, as the integral of a signed measure on [a,b].
What are rectifiable curves and how are they related to functions of bounded variation?
Rectifiable curves are curves that have finite length, which measures how long a curve is over an interval. The length of a curve C over an interval [a,b] is defined as follows:
The length of C over [a,b] is equal to the supremum of the sums of distances between consecutive points on C over all possible partitions of [a,b]. That is,
$$ L_a^b(C) = \sup \left\ \sum_i=1^n d(C(x_i), C(x_i-1)) : a = x_0 where d is the Euclidean distance and the supremum is taken over all possible choices of n and x_i for each i.
Rectifiable curves can be related to functions of bounded variation using parametrizations, arc length, and tangents. For example:
Parametrizations: A curve C can be parametrized by a function f from [a,b] to Euclidean space such that C(x) = f(x) for each x in [a,b]. If f is continuous and has bounded variation on [a,b], then C is rectifiable and its length is equal to the total variation of f.
Arc length: A curve C can be reparametrized by its arc length functi