The Lebesgue measure is an extension of the classical notions of length and area to more complicated sets. Given an open set containing disjoint intervals, the Lebesgue measure is defined byGiven a closed set ,A unit line segment has Lebesgue measure 1; the Cantor set has Lebesgue measure 0. The Minkowski measure of a bounded, closed set is the same as its Lebesgue measure (Ko 1995).
The Lebesgue integral is defined in terms of upper and lower bounds using the Lebesgue measure of a set. It uses a Lebesgue sum where is the value of the function in subinterval , and is the Lebesgue measure of the set of points for which values are approximately . This type of integral covers a wider class of functions than does the Riemann integral.The Lebesgue integral of a function over a measure space is writtenor sometimesto emphasize that the integral is taken with respect to the measure .
A nonnegative measurable function is called Lebesgue integrable if its Lebesgue integral is finite. An arbitrary measurable function is integrable if and are each Lebesgue integrable, where and denote the positive and negative parts of , respectively.The following equivalent characterization of Lebesgue integrable follows as a consequence of monotone convergence theorem. A nonnegative measurable function is Lebesgue integrable iff there exists a sequence of nonnegative simple functions such that the following two conditions are satisfied: 1. . 2. almost everywhere.
Suppose that is a sequence of measurable functions, that pointwise almost everywhere as , and that for all , where is integrable. Then is integrable, and
Any complex measure decomposes into an absolutely continuous measure and a singular measure , with respect to some positive measure . This is the Lebesgue decomposition,
The Darboux integral, also called a Darboux-Stieltjes integral, is a variant of the Stieltjes integral that is defined as a common value for the lower and upper Darboux integrals.Let and be bounded real functions on an interval , with nondecreasing. For any partition given by , let .The lower Darboux integral is the supremum of all lower sums of the formwhere denotes the infimum of over the interval .Likewise the upper Darboux integral is the infimum ofall upper sums of the formwhere denotes the supremum of over the interval .The lower Darboux integral is less or equal to the upper Darboux integral, and that the Darboux integral is a linear form on the vector space of Darboux-integrable functions on for a given .If , the original upper and lower Darboux integrals proposed by Darboux in 1875 are recovered.If the Stieltjes integral exists, then the Darboux integral also exists and has the same value. If is continuous, then the two integrals are identical...
Let and be measure spaces, let be the collection of all measurable rectangles contained in , and let be the premeasure defined on byfor . By the product measure , one means the Carathéodory extension of defined on the sigma-algebra of -measurable subsets of where denotes the outer measure induced by the premeasure on .
Let be a collection of subsets of a set and let be a set function. The function is called a premeasure provided that is finitely additive, countably monotone, and that if , where is the empty set.
A polar representation of a complex measure is analogous to the polar representation of a complex number as , where ,(1)The analog of absolute value is the total variation , and is replaced by a measurable real-valued function . Or sometimes one writes with instead of .More precisely, for any measurable set ,(2)where the integral is the Lebesgue integral. It is natural to extend the definition of the Lebesgue integral to complex measures using the polar representation(3)
If is a real measure (i.e., a measure that takes on real values), then one can decompose it according to where it is positive and negative. The positive variation is defined by(1)where is the total variation. Similarly, the negative variation is(2)Then the Jordan decomposition of is defined as(3)When already is a positive measure then . More generally, if is absolutely continuous, i.e.,(4)then so are and . The positive and negative variations can also be written as(5)and(6)where is the decomposition of into its positive and negative parts.The Jordan decomposition has a so-called minimum property. In particular, given any positive measure , the measure has another decomposition(7)The Jordan decomposition is minimal with respect to these changes. One way to say this is that any decomposition must have and ...
Define the correlation integral as(1)where is the Heaviside step function. When the below limit exists, the correlation dimension is then defined as(2)If is the correlation exponent, then(3)It satisfies(4)where is the capacity dimension and is the information dimension (correcting the typo in Baker and Gollub 1996), and is conjectured to be equal to the Lyapunov dimension.To estimate the correlation dimension of an -dimensional system with accuracy requires data points, where(5)where is the length of the "plateau region." If an attractor exists, then an estimate of saturates above some given by(6)which is sometimes known as the fractal Whitney embedding prevalence theorem.
The hypothesis is that, for is a measure space, for each , as . The hypothesis may be weakened to almost everywhere convergence.
Let be a bounded set in the plane, i.e., is contained entirely within a rectangle. The outer Jordan measure of is the greatest lower bound of the areas of the coverings of , consisting of finite unions of rectangles. The inner Jordan measure of is the difference between the area of an enclosing rectangle , and the outer measure of the complement of in . The Jordan measure, when it exists, is the common value of the outer and inner Jordan measures of .If is a bounded nonnegative function on the interval , the ordinate set of f is the setThen is Riemann integrable on iff is Jordan measurable, in which case the Jordan measure of is equal to .There are analogous versions of Jordan measure in all other dimensions.
A measure which takes values in the complex numbers. The set of complex measures on a measure space forms a vector space. Note that this is not the case for the more common positive measures. Also, the space of finite measures () has a norm given by the total variation measure , which makes it a Banach space.Using the polar representation of , it is possible to define the Lebesgue integral using a complex measure,Sometimes, the term "complex measure" is used to indicate an arbitrary measure. The definitions for measure can be extended to measures which take values in any vector space. For instance in spectral theory, measures on , which take values in the bounded linear maps from a Hilbert space to itself, represent the operator spectrum of an operator.
Let , then for any operator ,is called the Pincherle derivative of . If is a shift-invariant operator, then its Pincherle derivative is also a shift-invariant operator.
Measure theory is the study of measures. It generalizes the intuitive notions of length, area, and volume. The earliest and most important examples are Jordan measure and Lebesgue measure, but other examples are Borel measure, probability measure, complex measure, and Haar measure.
A function is Fréchet differentiable at ifexists. This is equivalent to the statement that has a removable discontinuity at , whereIn literature, the Fréchet derivative is sometimes known as the strong derivative (Ostaszewski 2012) and can be seen as a generalization of the gradient to arbitrary vector spaces (Long 2009).Every function which is Fréchet differentiable is both Carathéodory differentiable and Gâteaux differentiable. The relationship between the Fréchet derivative and the Gâteaux derivative can be made even more explicit by noting that a function is Fréchet differentiable if and only if the limit used to describe the Gâteaux derivative exists uniformly with respect to vectors on the unit sphere of the domain space ; as such, this uniform limit (when it exists) is what's called the Fréchet Derivative (Andrews and Hopper 2011)..
Given a complex measure , there exists a positive measure denoted which measures the total variation of , also sometimes called simply "total variation." In particular, on a subset is the largest sum of "variations" for any subdivision of . Roughly speaking, a total variation measure is an infinitesimal version of the absolute value.More precisely,(1)where the supremum is taken over all partitions of into measurable subsets .Note that may not be the same as . When already is a positive measure, then . More generally, if is absolutely continuous, that is(2)then so is , and the total variation measure can be written as(3)The total variation measure can be used to rewrite the original measure, in analogy to the norm of a complex number. The measure has a polar representation(4)with ...
The terms "measure," "measurable," etc. have very precise technical definitions (usually involving sigma-algebras) that can make them appear difficult to understand. However, the technical nature of the definitions is extremely important, since it gives a firm footing to concepts that are the basis for much of analysis (including some of the slippery underpinnings of calculus).For example, every definition of an integral is based on a particular measure: the Riemann integral is based on Jordan measure, and the Lebesgue integral is based on Lebesgue measure. The study of measures and their application to integration is known as measure theory.A measure is defined as a nonnegative real function from a delta-ring such that(1)where is the empty set, and(2)for any finite or countable collection of pairwise disjoint sets in such that is also in .If is -finite and is bounded, then can be extended uniquely to a measure defined..
If is square integrable over the real -axis, then any one of the following implies the other two: 1. The Fourier transform is 0 for . 2. Replacing by , the function is analytic in the complex plane for and approaches almost everywhere as . Furthermore, for some number and (i.e., the integral is bounded). 3. The real and imaginary parts of are Hilbert transforms of each other (Bracewell 1999, Problem 8, p. 273).
The Stieltjes integral is a generalization of the Riemann integral. Let and be real-valued bounded functions defined on a closed interval . Take a partition of the interval(1)and consider the Riemann sum(2)with . If the sum tends to a fixed number as , then is called the Stieltjes integral, or sometimes the Riemann-Stieltjes integral. The Stieltjes integral of with respect to is denoted(3)or sometimes simply(4)If and have a common point of discontinuity, then the integral does not exist. However, if is continuous and is Riemann integrable over the specified interval, then(5)(Kestelman 1960).For enumeration of many properties of the Stieltjes integral, see Dresher (1981, p. 105).
Let and be measure spaces. A measurable rectangle is a set of the form for and .
Let be Lebesgue integrable and let(1)be the corresponding Poisson integral. Then almost everywhere in (2)Let(3)be regular for , and let the integral(4)be bounded for . This condition is equivalent to the convergence of(5)Then almost everywhere in ,(6)Furthermore, is measurable, is Lebesgue integrable, and the Fourier series of is given by writing .
If is a sigma-algebra and is a subset of , then is called measurable if is a member of . need not have, a priori, a topological structure. Even if it does, there may be no connection between the open sets in the topology and the given sigma-algebra.
If is a sequence of nonnegative measurable functions, then(1)An example of a sequence of functions for which the inequality becomes strict is given by(2)
Two complex measures and on a measure space , are mutually singular if they are supported on different subsets. More precisely, where and are two disjoint sets such that the following hold for any measurable set , 1. The sets and are measurable. 2. The total variation of is supported on and that of on , i.e.,The relation of two measures being singular, written as , is plainly symmetric. Nevertheless, it is sometimes said that " is singular with respect to ."A discrete singular measure (with respect to Lebesgue measure on the reals) is a measure supported at 0, say iff . In general, a measure is concentrated on a subset if . For instance, the measure above is concentrated at 0.
A function is measurable if, for every real number , the setis measurable. When with Lebesgue measure, or more generally any Borel measure, then all continuous functions are measurable. In fact, practically any function that can be described is measurable. Measurable functions are closed under addition and multiplication, but not composition.The measurable functions form one of the most general classes of real functions. They are one of the basic objects of study in analysis, both because of their wide practical applicability and the aesthetic appeal of their generality. Whether a function is measurable depends on the measure on , and, in particular, it only depends on the sigma-algebra of measurable sets in . Sometimes, the measure on may be assumed to be a standard measure. For instance, a measurable function on is usually measurable with respect to Lebesgue measure.From the point of view of measure theory, subsets with measure zero do..
For a polynomial , the Mahler measure of is defined by(1)Using Jensen's formula, it can be shown that for ,(2)(Borwein and Erdélyi 1995, p. 271).Specific cases are given by(3)(4)(5)(Borwein and Erdélyi 1995, p. 272).A product of cyclotomic polynomials has Mahler measure 1. The Mahler measure of an integer polynomial in variables gives the topological entropy of a -dynamical system canonically associated to the polynomial.Lehmer's Mahler measure problem conjectures that a particular univariate polynomial has the smallest possible Mahler measure other than 1.
Euler integration was defined by Schanuel and subsequently explored by Rota, Chen, and Klain. The Euler integral of a function (assumed to be piecewise-constant with finitely many discontinuities) is the sum ofover the finitely many discontinuities of . The -dimensional Euler integral can be defined for classes of functions . Euler integration is additive, so the Euler integral of equals the sum of the Euler integrals of and .
A Cantor set in is said to be scrawny if for each neighborhood of an arbitrary point in , there is a neighborhood of such that every map extends to a map such that is finite. Babich (1992) presents examples of wild Cantor sets of this type and provides a proof that such objects cannot be defined by solid tori.
Let be a finite and measurable function in , and let be freely chosen. Then there is a function such that 1. is continuous in , 2. The measure of is , 3. , where denotes the upper bound of the set of the values of as runs through all values of .
The essential supremum is the proper generalization to measurable functions of the maximum. The technical difference is that the values of a function on a set of measure zero don't affect the essential supremum.Given a measurable function , where is a measure space with measure , the essential supremum is the smallest number such that the sethas measure zero. If no such number exists, as in the case of on , then the essential supremum is .The essential supremum of the absolute value of a function is usually denoted , and this serves as the norm for L-infty-space.
There are a couple of versions of this theorem. Basically, it says that any bounded linear functional on the space of compactly supported continuous functions on is the same as integration against a measure ,Here, the integral is the Lebesgue integral.Because linear functionals form a vector space, and are not "positive," the measure may not be a positive measure. But if the functional is positive, in the sense that implies that , then the measure is also positive. In the generality of complex linear functionals, the measure is a complex measure. The measure is uniquely determined by and has the properties of a regular Borel measure. It must be a finite measure, which corresponds to the boundedness condition on the functional. In fact, the operator norm of , , is the total variation measure of , .Naturally, there are some hypotheses necessary for this to make sense. The space has to be locally compact and a T2-Space, which is not a strong..
where is the measure of the set of points on the x-axis for which .
The term energy has an important physical meaning in physics and is an extremely useful concept. A much more abstract mathematical generalization is defined as follows. Let be a space with measure and let be a real function on the product space . When (1)(2)exists for measures , is called the mutual energy and is called the energy.
Let be a measure space and let be a measurable set with . Let be a sequence of measurable functions on such that each is finite almost everywhere in and converges almost everywhere in to a finite limit. Then for every , there exists a subset of with such that converges uniformly on .If and is either the class of Borel sets or the class of Lebesgue measurable sets, then the set can be chosen to be a closed set.
A set function is said to possess countable subadditivity if, given any countable disjoint collection of sets on which is defined,A function possessing countable subadditivity is said to be countably subadditive.Any countably subadditive function is also finitely subadditive presuming that where is the empty set.
Let be a set and a collection of subsets of . A set function is said to possess countable monotonicity provided that, whenever a set is covered by a countable collection of sets in ,A function which possesses countable monotonicity issaid to be countably monotone.One can easily verify that any set function which is both monotone (in the sense of mapping subsets of the domain to subsets of the range) and countably additive is necessarily countably monotone. The converse is not true in general.
A set function is said to possess finite subadditivity if, given any finite disjoint collection of sets on which is defined,A set function possessing finite subadditivity is said to be finitely subadditive. In particular, every finitely additive set function is also finitely subadditive.
A set function possesses countable additivity if, given any countable disjoint collection of sets on which is defined,A function having countable additivity is said to becountably additive.Countably additive functions are countably subadditive by definition. Moreover, provided that where is the empty set, every countably additive function is necessarily finitely additive.
Let be a set and a collection of subsets of . A set function is said to possess finite monotonicity provided that, whenever a set is covered by a finite collection of sets in ,A set function possessing finite monotonicity is said to be finitely monotone. Note that a set function which is countably monotone is necessarily finitely monotone provided that and , where is the empty set.
A set function is finitely additive if, given any finite disjoint collection of sets on which is defined,
The Euclidean metric is the function that assigns to any two vectors in Euclidean -space and the number(1)and so gives the "standard" distance between any two vectors in .The Euclidean metric in Euclidean three-space is given by(2)giving the line element(3)(4)where Einstein summation has been used.
Let be a collection of subsets of a set , a set function, and the outer measure induced by . The measure that is the restriction of to the sigma-algebra of -measurable sets is called the Carathéodory measure induced by .Perhaps somewhat surprisingly, even though is a measure induced by the set function , it may not be the case that is an extension of . In the event that does extend , is called the Carathéodory extension of .
Given a set , a set function is said to be an outer measure provided that and that is countably monotone, where is the empty set.Given a collection of subsets of and an arbitrary set function , one can define a new set function by setting and defining, for each non-empty subset ,where the infimum is taken over all countable collections of sets in which cover . The resulting function is an outer measure and is called the outer measure induced by .
A function is Carathéodory differentiable at if there exists a function which is continuous at such thatEvery function which is Carathéodory differentiable is also Fréchetdifferentiable.
Let be a metric space, be a subset of , and a number . The -dimensional Hausdorff measure of , , is the infimum of positive numbers such that for every , can be covered by a countable family of closed sets, each of diameter less than , such that the sum of the th powers of their diameters is less than . Note that may be infinite, and need not be an integer.
Let be a space with measure , and let be a real function on the product space . When(1)(2)exists for measures , is called the mutual energy. is then called the energy.
The standard Gauss measure of a finite dimensional real Hilbert space with norm has the Borel measurewhere is the Lebesgue measure on .
A measure is absolutely continuous with respect to another measure if for every set with . This makes sense as long as is a positive measure, such as Lebesgue measure, but can be any measure, possibly a complex measure.By the Radon-Nikodym theorem, this is equivalentto saying thatwhere the integral is the Lebesgue integral, for some integrable function . The function is like a derivative, and is called the Radon-Nikodym derivative .The measure supported at 0 ( iff ) is not absolutely continuous with respect to Lebesgue measure, and is a singular measure.
The French metro metric is an example for disproving apparently intuitive but false properties of metric spaces. The metric consists of a distance function on the plane such that for all ,(1)where is the normal distance function on the plane. This metric has the property that for , the open ball of radius around is an open line segment along vector , while for , the open ball is the union of a line segment and an open disk around the origin.