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Measure (mathematics)

From Wikipedia, the free encyclopedia
Generalization of mass, length, area and volume
For the coalgebraic concept, seeMeasuring coalgebra.
Not to be confused withMetric (mathematics).
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Informally, a measure has the property of beingmonotone in the sense that ifA{\displaystyle A} is asubset ofB,{\displaystyle B,} the measure ofA{\displaystyle A} is less than or equal to the measure ofB.{\displaystyle B.} Furthermore, the measure of theempty set is required to be 0. A simple example is a volume (how big an object occupies a space) as a measure.

Inmathematics, the concept of ameasure is a generalization and formalization ofgeometrical measures (length,area,volume) and other common notions, such asmagnitude,mass, andprobability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational inprobability theory,integration theory, and can be generalized to assumenegative values, as withelectrical charge. Far-reaching generalizations (such asspectral measures andprojection-valued measures) of measure are widely used inquantum physics and physics in general.

The intuition behind this concept dates back toAncient Greece, whenArchimedes tried to calculate thearea of a circle.[1][2] But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works ofÉmile Borel,Henri Lebesgue,Nikolai Luzin,Johann Radon,Constantin Carathéodory, andMaurice Fréchet, among others.

Definition

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Countable additivity of a measureμ{\displaystyle \mu }: The measure of a countable disjoint union is the same as the sum of all measures of each subset.

LetX{\displaystyle X} be a set andΣ{\displaystyle \Sigma } aσ-algebra overX.{\displaystyle X.} Aset functionμ{\displaystyle \mu } fromΣ{\displaystyle \Sigma } to theextended real number line is called ameasure if the following conditions hold:

If at least one setE{\displaystyle E} has finite measure, then the requirementμ(∅)=0{\displaystyle \mu (\varnothing )=0} is met automatically due to countable additivity:μ(E)=μ(E∪∅)=μ(E)+μ(∅),{\displaystyle \mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing ),}and thereforeμ(∅)=0.{\displaystyle \mu (\varnothing )=0.}

If the condition of non-negativity is dropped, andμ{\displaystyle \mu } takes on at most one of the values of±∞,{\displaystyle \pm \infty ,} thenμ{\displaystyle \mu } is called asigned measure.

The pair(X,Σ){\displaystyle (X,\Sigma )} is called ameasurable space, and the members ofΣ{\displaystyle \Sigma } are calledmeasurable sets.

Atriple(X,Σ,μ){\displaystyle (X,\Sigma ,\mu )} is called ameasure space. Aprobability measure is a measure with total measure one – that is,μ(X)=1.{\displaystyle \mu (X)=1.} Aprobability space is a measure space with a probability measure.

For measure spaces that are alsotopological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice inanalysis (and in many cases also inprobability theory) areRadon measures. Radon measures have an alternative definition in terms of linear functionals on thelocally convex topological vector space ofcontinuous functions withcompact support. This approach is taken byBourbaki (2004) and a number of other sources. For more details, see the article onRadon measures.

Instances

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Main category:Measures (measure theory)

Some important measures are listed here.

Other 'named' measures used in various theories include:Borel measure,Jordan measure,ergodic measure,Gaussian measure,Baire measure,Radon measure,Young measure, andLoeb measure.

In physics an example of a measure is spatial distribution ofmass (see for example,gravity potential), or another non-negativeextensive property,conserved (seeconservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.

Measure theory is used in machine learning. One example is the Flow Induced Probability Measure in GFlowNet.[3]

Basic properties

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Letμ{\displaystyle \mu } be a measure.

Monotonicity

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IfE1{\displaystyle E_{1}} andE2{\displaystyle E_{2}} are measurable sets withE1⊆E2{\displaystyle E_{1}\subseteq E_{2}} thenμ(E1)≤μ(E2).{\displaystyle \mu (E_{1})\leq \mu (E_{2}).}

Measure of countable unions and intersections

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Countable subadditivity

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For anycountablesequenceE1,E2,E3,…{\displaystyle E_{1},E_{2},E_{3},\ldots } of (not necessarily disjoint) measurable setsEn{\displaystyle E_{n}} inΣ:{\displaystyle \Sigma :}μ(⋃i=1∞Ei)≤∑i=1∞μ(Ei).{\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mu (E_{i}).}

Continuity from below

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IfE1,E2,E3,…{\displaystyle E_{1},E_{2},E_{3},\ldots } are measurable sets that are increasing (meaning thatE1⊆E2⊆E3⊆…{\displaystyle E_{1}\subseteq E_{2}\subseteq E_{3}\subseteq \ldots }) then theunion of the setsEn{\displaystyle E_{n}} is measurable andμ(⋃i=1∞Ei) = limi→∞μ(Ei)=supi≥1μ(Ei).{\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)~=~\lim _{i\to \infty }\mu (E_{i})=\sup _{i\geq 1}\mu (E_{i}).}

Continuity from above

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IfE1,E2,E3,…{\displaystyle E_{1},E_{2},E_{3},\ldots } are measurable sets that are decreasing (meaning thatE1⊇E2⊇E3⊇…{\displaystyle E_{1}\supseteq E_{2}\supseteq E_{3}\supseteq \ldots }) then theintersection of the setsEn{\displaystyle E_{n}} is measurable; furthermore, if at least one of theEn{\displaystyle E_{n}} has finite measure thenμ(⋂i=1∞Ei)=limi→∞μ(Ei)=infi≥1μ(Ei).{\displaystyle \mu \left(\bigcap _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})=\inf _{i\geq 1}\mu (E_{i}).}

This property is false without the assumption that at least one of theEn{\displaystyle E_{n}} has finite measure. For instance, for eachn∈N,{\displaystyle n\in \mathbb {N} ,} letEn=[n,∞)⊆R,{\displaystyle E_{n}=[n,\infty )\subseteq \mathbb {R} ,} which all have infinite Lebesgue measure, but the intersection is empty.

Other properties

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Completeness

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Main article:Complete measure

A measurable setX{\displaystyle X} is called anull set ifμ(X)=0.{\displaystyle \mu (X)=0.} A subset of a null set is called anegligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is calledcomplete if every negligible set is measurable.

A measure can be extended to a complete one by considering the σ-algebra of subsetsY{\displaystyle Y} which differ by a negligible set from a measurable setX,{\displaystyle X,} that is, such that thesymmetric difference ofX{\displaystyle X} andY{\displaystyle Y} is contained in a null set. One definesμ(Y){\displaystyle \mu (Y)} to equalμ(X).{\displaystyle \mu (X).}

"Dropping the Edge"

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Iff:X→[0,+∞]{\displaystyle f:X\to [0,+\infty ]} is(Σ,B([0,+∞])){\displaystyle (\Sigma ,{\cal {B}}([0,+\infty ]))}-measurable, thenμ{x∈X:f(x)≥t}=μ{x∈X:f(x)>t}{\displaystyle \mu \{x\in X:f(x)\geq t\}=\mu \{x\in X:f(x)>t\}}foralmost allt∈[−∞,∞].{\displaystyle t\in [-\infty ,\infty ].}[4] This property is used in connection withLebesgue integral.

Proof

BothF(t):=μ{x∈X:f(x)>t}{\displaystyle F(t):=\mu \{x\in X:f(x)>t\}} andG(t):=μ{x∈X:f(x)≥t}{\displaystyle G(t):=\mu \{x\in X:f(x)\geq t\}} are monotonically non-increasing functions oft,{\displaystyle t,} so both of them haveat most countably many discontinuities and thus they are continuous almost everywhere, relative to the Lebesgue measure. Ift<0{\displaystyle t<0} then{x∈X:f(x)≥t}=X={x∈X:f(x)>t},{\displaystyle \{x\in X:f(x)\geq t\}=X=\{x\in X:f(x)>t\},} so thatF(t)=G(t),{\displaystyle F(t)=G(t),} as desired.

Ift{\displaystyle t} is such thatμ{x∈X:f(x)>t}=+∞{\displaystyle \mu \{x\in X:f(x)>t\}=+\infty } thenmonotonicity impliesμ{x∈X:f(x)≥t}=+∞,{\displaystyle \mu \{x\in X:f(x)\geq t\}=+\infty ,}so thatF(t)=G(t),{\displaystyle F(t)=G(t),} as required. Ifμ{x∈X:f(x)>t}=+∞{\displaystyle \mu \{x\in X:f(x)>t\}=+\infty } for allt{\displaystyle t} then we are done, so assume otherwise. Then there is a uniquet0∈{−∞}∪[0,+∞){\displaystyle t_{0}\in \{-\infty \}\cup [0,+\infty )} such thatF{\displaystyle F} is infinite to the left oft{\displaystyle t} (which can only happen whent0≥0{\displaystyle t_{0}\geq 0}) and finite to the right. Arguing as above,μ{x∈X:f(x)≥t}=+∞{\displaystyle \mu \{x\in X:f(x)\geq t\}=+\infty } whent<t0.{\displaystyle t<t_{0}.} Similarly, ift0≥0{\displaystyle t_{0}\geq 0} andF(t0)=+∞{\displaystyle F\left(t_{0}\right)=+\infty } thenF(t0)=G(t0).{\displaystyle F\left(t_{0}\right)=G\left(t_{0}\right).}

Fort>t0,{\displaystyle t>t_{0},} lettn{\displaystyle t_{n}} be a monotonically non-decreasing sequence converging tot.{\displaystyle t.} The monotonically non-increasing sequences{x∈X:f(x)>tn}{\displaystyle \{x\in X:f(x)>t_{n}\}} of members ofΣ{\displaystyle \Sigma } has at least one finitelyμ{\displaystyle \mu }-measurable component, and{x∈X:f(x)≥t}=⋂n{x∈X:f(x)>tn}.{\displaystyle \{x\in X:f(x)\geq t\}=\bigcap _{n}\{x\in X:f(x)>t_{n}\}.}Continuity from above guarantees thatμ{x∈X:f(x)≥t}=limtn↑tμ{x∈X:f(x)>tn}.{\displaystyle \mu \{x\in X:f(x)\geq t\}=\lim _{t_{n}\uparrow t}\mu \{x\in X:f(x)>t_{n}\}.}The right-hand sidelimtn↑tF(tn){\displaystyle \lim _{t_{n}\uparrow t}F\left(t_{n}\right)} then equalsF(t)=μ{x∈X:f(x)>t}{\displaystyle F(t)=\mu \{x\in X:f(x)>t\}} ift{\displaystyle t} is a point of continuity ofF.{\displaystyle F.} SinceF{\displaystyle F} is continuous almost everywhere, this completes the proof.

Additivity

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Measures are required to be countably additive. However, the condition can be strengthened as follows.For any setI{\displaystyle I} and any set of nonnegativeri,i∈I{\displaystyle r_{i},i\in I} define:∑i∈Iri=sup{∑i∈Jri:|J|<∞,J⊆I}.{\displaystyle \sum _{i\in I}r_{i}=\sup \left\lbrace \sum _{i\in J}r_{i}:|J|<\infty ,J\subseteq I\right\rbrace .}That is, we define the sum of theri{\displaystyle r_{i}} to be the supremum of all the sums of finitely many of them.

A measureμ{\displaystyle \mu } onΣ{\displaystyle \Sigma } isκ{\displaystyle \kappa }-additive if for anyλ<κ{\displaystyle \lambda <\kappa } and any family of disjoint setsXα,α<λ{\displaystyle X_{\alpha },\alpha <\lambda } the following hold:⋃α∈λXα∈Σ{\displaystyle \bigcup _{\alpha \in \lambda }X_{\alpha }\in \Sigma }μ(⋃α∈λXα)=∑α∈λμ(Xα).{\displaystyle \mu \left(\bigcup _{\alpha \in \lambda }X_{\alpha }\right)=\sum _{\alpha \in \lambda }\mu \left(X_{\alpha }\right).}The second condition is equivalent to the statement that theideal of null sets isκ{\displaystyle \kappa }-complete.

Sigma-finite measures

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Main article:Sigma-finite measure

A measure space(X,Σ,μ){\displaystyle (X,\Sigma ,\mu )} is called finite ifμ(X){\displaystyle \mu (X)} is a finite real number (rather than∞{\displaystyle \infty }). Nonzero finite measures are analogous toprobability measures in the sense that any finite measureμ{\displaystyle \mu } is proportional to the probability measure1μ(X)μ.{\displaystyle {\frac {1}{\mu (X)}}\mu .} A measureμ{\displaystyle \mu } is calledσ-finite ifX{\displaystyle X} can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have aσ-finite measure if it is a countable union of sets with finite measure.

For example, thereal numbers with the standardLebesgue measure are σ-finite but not finite. Consider theclosed intervals[k,k+1]{\displaystyle [k,k+1]} for allintegersk;{\displaystyle k;} there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider thereal numbers with thecounting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to theLindelöf property of topological spaces.[original research?] They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.

Strictly localizable measures

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Main article:Decomposable measure

Semifinite measures

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LetX{\displaystyle X} be a set, letA{\displaystyle {\cal {A}}} be a sigma-algebra onX,{\displaystyle X,} and letμ{\displaystyle \mu } be a measure onA.{\displaystyle {\cal {A}}.} We sayμ{\displaystyle \mu } issemifinite to mean that for allA∈μpre{+∞},{\displaystyle A\in \mu ^{\text{pre}}\{+\infty \},}P(A)∩μpre(R>0)≠∅.{\displaystyle {\cal {P}}(A)\cap \mu ^{\text{pre}}(\mathbb {R} _{>0})\neq \emptyset .}[5]

Semifinite measures generalize sigma-finite measures, in such a way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.)

Basic examples

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Involved example

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The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal toμ.{\displaystyle \mu .} It can be shown there is a greatest measure with these two properties:

Theorem (semifinite part)[9]—For any measureμ{\displaystyle \mu } onA,{\displaystyle {\cal {A}},} there exists, among semifinite measures onA{\displaystyle {\cal {A}}} that are less than or equal toμ,{\displaystyle \mu ,} agreatest elementμsf.{\displaystyle \mu _{\text{sf}}.}

We say thesemifinite part ofμ{\displaystyle \mu } to mean the semifinite measureμsf{\displaystyle \mu _{\text{sf}}} defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for the semifinite part:

Sinceμsf{\displaystyle \mu _{\text{sf}}} is semifinite, it follows that ifμ=μsf{\displaystyle \mu =\mu _{\text{sf}}} thenμ{\displaystyle \mu } is semifinite. It is also evident that ifμ{\displaystyle \mu } is semifinite thenμ=μsf.{\displaystyle \mu =\mu _{\text{sf}}.}

Non-examples

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Every0−∞{\displaystyle 0-\infty } measure that is not the zero measure is not semifinite. (Here, we say0−∞{\displaystyle 0-\infty } measure to mean a measure whose range lies in{0,+∞}{\displaystyle \{0,+\infty \}}:(∀A∈A)(μ(A)∈{0,+∞}).{\displaystyle (\forall A\in {\cal {A}})(\mu (A)\in \{0,+\infty \}).}) Below we give examples of0−∞{\displaystyle 0-\infty } measures that are not zero measures.

Involved non-example

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Measures that are not semifinite are very wild when restricted to certain sets.[Note 1] Every measure is, in a sense, semifinite once its0−∞{\displaystyle 0-\infty } part (the wild part) is taken away.

— A. Mukherjea and K. Pothoven,Real and Functional Analysis, Part A: Real Analysis (1985)

Theorem (Luther decomposition)[14][15]—For any measureμ{\displaystyle \mu } onA,{\displaystyle {\cal {A}},} there exists a0−∞{\displaystyle 0-\infty } measureξ{\displaystyle \xi } onA{\displaystyle {\cal {A}}} such thatμ=ν+ξ{\displaystyle \mu =\nu +\xi } for some semifinite measureν{\displaystyle \nu } onA.{\displaystyle {\cal {A}}.} In fact, among such measuresξ,{\displaystyle \xi ,} there exists aleast measureμ0−∞.{\displaystyle \mu _{0-\infty }.} Also, we haveμ=μsf+μ0−∞.{\displaystyle \mu =\mu _{\text{sf}}+\mu _{0-\infty }.}

We say the0−∞{\displaystyle \mathbf {0-\infty } } part ofμ{\displaystyle \mu } to mean the measureμ0−∞{\displaystyle \mu _{0-\infty }} defined in the above theorem. Here is an explicit formula forμ0−∞{\displaystyle \mu _{0-\infty }}:μ0−∞=(sup{μ(B)−μsf(B):B∈P(A)∩μsfpre(R≥0)})A∈A.{\displaystyle \mu _{0-\infty }=(\sup\{\mu (B)-\mu _{\text{sf}}(B):B\in {\cal {P}}(A)\cap \mu _{\text{sf}}^{\text{pre}}(\mathbb {R} _{\geq 0})\})_{A\in {\cal {A}}}.}

Results regarding semifinite measures

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Localizable measures

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Localizable measures are a special case of semifinite measures and a generalization of sigma-finite measures.

LetX{\displaystyle X} be a set, letA{\displaystyle {\cal {A}}} be a sigma-algebra onX,{\displaystyle X,} and letμ{\displaystyle \mu } be a measure onA.{\displaystyle {\cal {A}}.}

s-finite measures

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Main article:s-finite measure

A measure is said to be s-finite if it is a countable sum of finite measures. S-finite measures are more general than sigma-finite ones and have applications in the theory ofstochastic processes.

Non-measurable sets

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Main article:Non-measurable set

If theaxiom of choice is assumed to be true, it can be proved that not all subsets ofEuclidean space areLebesgue measurable; examples of such sets include theVitali set, and the non-measurable sets postulated by theHausdorff paradox and theBanach–Tarski paradox.

Generalizations

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For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additiveset function with values in the (signed) real numbers is called asigned measure, while such a function with values in thecomplex numbers is called acomplex measure. Observe, however, that complex measure is necessarily of finitevariation, hence complex measures includefinite signed measures but not, for example, theLebesgue measure.

Measures that take values inBanach spaces have been studied extensively.[22] A measure that takes values in the set of self-adjoint projections on aHilbert space is called aprojection-valued measure; these are used infunctional analysis for thespectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the termpositive measure is used. Positive measures are closed underconical combination but not generallinear combination, while signed measures are the linear closure of positive measures. More generally seemeasure theory in topological vector spaces.

Another generalization is thefinitely additive measure, also known as acontent. This is the same as a measure except that instead of requiringcountable additivity we require onlyfinite additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected with notions such asBanach limits, the dual ofL∞{\displaystyle L^{\infty }} and theStone–Čech compactification. All these are linked in one way or another to theaxiom of choice. Contents remain useful in certain technical problems ingeometric measure theory; this is the theory ofBanach measures.

Acharge is a generalization in both directions: it is a finitely additive, signed measure.[23] (Cf.ba space for information aboutbounded charges, where we say a charge isbounded to mean its range its a bounded subset ofR.)

See also

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Notes

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  1. ^One way to rephrase our definition is thatμ{\displaystyle \mu } is semifinite if and only if(∀A∈μpre{+∞})(∃B⊆A)(0<μ(B)<+∞).{\displaystyle (\forall A\in \mu ^{\text{pre}}\{+\infty \})(\exists B\subseteq A)(0<\mu (B)<+\infty ).} Negating this rephrasing, we find thatμ{\displaystyle \mu } is not semifinite if and only if(∃A∈μpre{+∞})(∀B⊆A)(μ(B)∈{0,+∞}).{\displaystyle (\exists A\in \mu ^{\text{pre}}\{+\infty \})(\forall B\subseteq A)(\mu (B)\in \{0,+\infty \}).} For every such setA,{\displaystyle A,} the subspace measure induced by the subspace sigma-algebra induced byA,{\displaystyle A,} i.e. the restriction ofμ{\displaystyle \mu } to said subspace sigma-algebra, is a0−∞{\displaystyle 0-\infty } measure that is not the zero measure.

Bibliography

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References

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  1. ^ArchimedesMeasuring the Circle
  2. ^Heath, T. L. (1897). "Measurement of a Circle".The Works Of Archimedes. Osmania University, Digital Library Of India. Cambridge University Press. pp. 91–98.
  3. ^Bengio, Yoshua; Lahlou, Salem; Deleu, Tristan; Hu, Edward J.; Tiwari, Mo; Bengio, Emmanuel (2021). "GFlowNet Foundations".arXiv:2111.09266 [cs.LG].
  4. ^Fremlin, D. H. (2010),Measure Theory, vol. 2 (Second ed.), p. 221
  5. ^abcMukherjea & Pothoven 1985, p. 90.
  6. ^Folland 1999, p. 25.
  7. ^Edgar 1998, Theorem 1.5.2, p. 42.
  8. ^Edgar 1998, Theorem 1.5.3, p. 42.
  9. ^abNielsen 1997, Exercise 11.30, p. 159.
  10. ^Fremlin 2016, Section 213X, part (c).
  11. ^Royden & Fitzpatrick 2010, Exercise 17.8, p. 342.
  12. ^Hewitt & Stromberg 1965, part (b) of Example 10.4, p. 127.
  13. ^Fremlin 2016, Section 211O, p. 15.
  14. ^abLuther 1967, Theorem 1.
  15. ^Mukherjea & Pothoven 1985, part (b) of Proposition 2.3, p. 90.
  16. ^Fremlin 2016, part (a) of Theorem 243G, p. 159.
  17. ^abFremlin 2016, Section 243K, p. 162.
  18. ^Fremlin 2016, part (a) of the Theorem in Section 245E, p. 182.
  19. ^Fremlin 2016, Section 245M, p. 188.
  20. ^Berberian 1965, Theorem 39.1, p. 129.
  21. ^Fremlin 2016, part (b) of Theorem 243G, p. 159.
  22. ^Rao, M. M. (2012),Random and Vector Measures, Series on Multivariate Analysis, vol. 9,World Scientific,ISBN 978-981-4350-81-5,MR 2840012.
  23. ^Bhaskara Rao, K. P. S. (1983).Theory of charges: a study of finitely additive measures. M. Bhaskara Rao. London: Academic Press. p. 35.ISBN 0-12-095780-9.OCLC 21196971.
  24. ^Folland 1999, p. 27, Exercise 1.15.a.

External links

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Look upmeasurable in Wiktionary, the free dictionary.
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