View source for Measure (mathematics)

{{short description|Generalization of mass, length, area and volume}}
{{For|the coalgebraic concept|Measuring coalgebra}}
{{Distinguish|Metric (mathematics)}}
{{More footnotes|date=January 2021}}
[[File:Measure illustration (Vector).svg|alt=|thumb|Informally, a measure has the property of being [[Monotone function|monotone]] in the sense that if <math>A</math> is a [[subset]] of <math>B,</math> the measure of <math>A</math> is less than or equal to the measure of <math>B.</math> Furthermore, the measure of the [[empty set]] is required to be 0. A simple example is a volume (how big an object occupies a space) as a measure.]]

In [[mathematics]], the concept of a '''measure''' is a generalization and formalization of [[geometrical measures]] ([[length]], [[area]], [[volume]]) and other common notions, such as [[magnitude (mathematics)|magnitude]], [[mass]], and [[probability]] of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in [[probability theory]], [[integral|integration theory]], and can be generalized to assume [[signed measure|negative values]], as with [[electrical charge]]. Far-reaching generalizations (such as [[spectral measure]]s and [[projection-valued measure]]s) of measure are widely used in [[quantum physics]] and physics in general.

The intuition behind this concept dates back to [[ancient Greece]], when [[Archimedes]] tried to calculate the [[area of a circle]].<ref>Archimedes [https://web.archive.org/web/20040703122928/http://www.math.ubc.ca/~cass/archimedes/circle.html Measuring the Circle]</ref><ref>{{Cite book |last=Heath |first=T. L. |url=http://archive.org/details/worksofarchimede029517mbp |title=The Works Of Archimedes |date=1897 |publisher=Cambridge University Press. |others=Osmania University, Digital Library Of India |pages=91–98 |chapter=Measurement of a Circle}}</ref> 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 of [[Émile Borel]], [[Henri Lebesgue]], [[Nikolai Luzin]], [[Johann Radon]], [[Constantin Carathéodory]], and [[Maurice Fréchet]], among others.

==Definition==
[[File:Countable additivity of a measure.svg|thumb|300px|Countable additivity of a measure <math>\mu</math>: The measure of a countable disjoint union is the same as the sum of all measures of each subset.]]

Let <math>X</math> be a set and <math>\Sigma</math> a [[σ-algebra]] over <math>X.</math> A [[set function]] <math>\mu</math> from <math>\Sigma</math> to the [[extended real number line]] is called a '''measure''' if the following conditions hold:

*'''Non-negativity''': For all <math>E \in \Sigma, \ \ \mu(E) \geq 0.</math>
*<math>\mu(\varnothing) = 0.</math>
*'''Countable additivity''' (or [[sigma additivity|σ-additivity]]): For all [[countable]] collections <math>\{ E_k \}_{k=1}^\infty</math> of pairwise [[disjoint sets]] in Σ,<math display="block">\mu{\left(\bigcup_{k=1}^\infty E_k\right)} = \sum_{k=1}^\infty \mu(E_k).</math>

If at least one set <math>E</math> has finite measure, then the requirement <math>\mu(\varnothing) = 0</math> is met automatically due to countable additivity:
<math display=block>\mu(E)=\mu(E \cup \varnothing) = \mu(E) + \mu(\varnothing),</math>
and therefore <math>\mu(\varnothing)=0.</math>

If the condition of non-negativity is dropped, and <math>\mu</math> takes on at most one of the values of <math>\pm \infty,</math> then <math>\mu</math> is called a ''[[signed measure]]''.

The pair <math>(X, \Sigma)</math> is called a ''[[measurable space]]'', and the members of <math>\Sigma</math> are called '''measurable sets'''.

A [[tuple|triple]] <math>(X, \Sigma, \mu)</math> is called a ''[[measure space]]''. A [[probability measure]] is a measure with total measure one – that is, <math>\mu(X) = 1.</math> A [[probability space]] is a measure space with a probability measure.

For measure spaces that are also [[topological space]]s various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in [[analysis (mathematics)|analysis]] (and in many cases also in [[probability theory]]) are [[Radon measure]]s. Radon measures have an alternative definition in terms of linear functionals on the [[locally convex topological vector space]] of [[continuous function]]s with [[Support (mathematics)#Compact support|compact support]]. This approach is taken by [[Nicolas Bourbaki|Bourbaki]] (2004) and a number of other sources. For more details, see the article on [[Radon measure]]s.

==Instances==
{{main category|Measures (measure theory)}}
Some important measures are listed here.

* The [[counting measure]] is defined by <math>\mu(S)</math> = number of elements in <math>S.</math>
* The [[Lebesgue measure]] on <math>\R</math> is a [[Complete measure|complete]] [[translational invariance|translation-invariant]] measure on a ''σ''-algebra containing the [[interval (mathematics)|interval]]s in <math>\R</math> such that <math>\mu([0, 1]) = 1</math>; and every other measure with these properties extends the Lebesgue measure.
* Circular [[angle]] measure is invariant under [[rotation]], and [[hyperbolic angle]] measure is invariant under [[squeeze mapping]].
* The [[Haar measure]] for a [[Locally compact space|locally compact]] [[topological group]] is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.
*Every (pseudo) [[Riemannian manifold]] <math>(M,g)</math> has a canonical measure <math>\mu_g</math> that in local coordinates <math>x_1,\ldots,x_n</math> looks like <math>\sqrt{\left|\det g \right|}d^nx</math> where <math>d^nx</math> is the usual Lebesgue measure.
* The [[Hausdorff measure]] is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets.
* Every [[probability space]] gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the [[unit interval]] [0, 1]). Such a measure is called a ''probability measure'' or ''distribution''. See the [[list of probability distributions]] for instances.
* The [[Dirac measure]] ''δ''<sub>''a''</sub> (cf. [[Dirac delta function]]) is given by ''δ''<sub>''a''</sub>(''S'') = ''χ''<sub>''S''</sub>(a), where ''χ''<sub>''S''</sub> is the [[indicator function]] of <math>S.</math> The measure of a set is 1 if it contains the point <math>a</math> and 0 otherwise.

Other 'named' measures used in various theories include: [[Borel measure]], [[Jordan measure]], [[ergodic measure]], [[Gaussian measure]], [[Baire measure]], [[Radon measure]], [[Young measure]], and [[Loeb measure]].
 
In physics an example of a measure is spatial distribution of [[mass]] (see for example, [[gravity potential]]), or another non-negative [[extensive property]], [[conserved quantity|conserved]] (see [[Conservation law (physics)|conservation law]] for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.

* [[Liouville's theorem (Hamiltonian)#Symplectic geometry|Liouville measure]], known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.
* [[Gibbs measure]] is widely used in statistical mechanics, often under the name [[canonical ensemble]].

Measure theory is used in machine learning. One example is the Flow Induced Probability Measure in GFlowNet.<ref>{{cite arXiv | eprint=2111.09266 | last1=Bengio | first1=Yoshua | last2=Lahlou | first2=Salem | last3=Deleu | first3=Tristan | last4=Hu | first4=Edward J. | last5=Tiwari | first5=Mo | last6=Bengio | first6=Emmanuel | title=GFlowNet Foundations | date=2021 | class=cs.LG }}</ref>

==Basic properties==
Let <math>\mu</math> be a measure.

===Monotonicity===
If <math>E_1</math> and <math>E_2</math> are measurable sets with <math>E_1 \subseteq E_2</math> then
<math display=block>\mu(E_1) \leq \mu(E_2).</math>

===Measure of countable unions and intersections===
====Countable subadditivity====
For any [[countable]] [[Sequence (mathematics)|sequence]] <math>E_1, E_2, E_3, \ldots</math> of (not necessarily disjoint) measurable sets <math>E_n</math> in <math>\Sigma:</math>
<math display=block>\mu\left( \bigcup_{i=1}^\infty E_i\right) \leq \sum_{i=1}^\infty \mu(E_i).</math>

====Continuity from below====
If <math>E_1, E_2, E_3, \ldots</math> are measurable sets that are increasing (meaning that <math>E_1 \subseteq E_2 \subseteq E_3 \subseteq \ldots</math>) then the [[Union (set theory)|union]] of the sets <math>E_n</math> is measurable and
<math display=block>\mu\left(\bigcup_{i=1}^\infty E_i\right) ~=~ \lim_{i\to\infty}  \mu(E_i) = \sup_{i \geq 1} \mu(E_i).</math>

====Continuity from above====
If <math>E_1, E_2, E_3, \ldots</math> are measurable sets that are decreasing (meaning that <math>E_1 \supseteq E_2 \supseteq E_3 \supseteq \ldots</math>) then the [[Intersection (set theory)|intersection]] of the sets <math>E_n</math> is measurable; furthermore, if at least one of the <math>E_n</math> has finite measure then
<math display=block>\mu\left(\bigcap_{i=1}^\infty E_i\right) = \lim_{i\to\infty} \mu(E_i) = \inf_{i \geq 1} \mu(E_i).</math>

This property is false without the assumption that at least one of the <math>E_n</math> has finite measure. For instance, for each <math>n \in \N,</math> let <math>E_n = [n, \infty) \subseteq \R,</math> which all have infinite Lebesgue measure, but the intersection is empty.

==Other properties==

===Completeness===
{{Main|Complete measure}}

A measurable set <math>X</math> is called a ''[[null set]]'' if <math>\mu(X) = 0.</math> A subset of a null set is called a ''negligible set''. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called ''complete'' if every negligible set is measurable.

A measure can be extended to a complete one by considering the σ-algebra of subsets <math>Y</math> which differ by a negligible set from a measurable set <math>X,</math> that is, such that the [[symmetric difference]] of <math>X</math> and <math>Y</math> is contained in a null set. One defines <math>\mu(Y)</math> to equal <math>\mu(X).</math>

==="Dropping the Edge"===
If <math>f:X\to[0,+\infty]</math> is <math>(\Sigma,{\cal B}([0,+\infty]))</math>-measurable, then
<math display=block>\mu\{x\in X: f(x) \geq t\} = \mu\{x\in X: f(x) > t\}</math>
for [[almost everywhere|almost all]] <math>t \in [-\infty,\infty].</math><ref>{{citation | last = Fremlin | first = D. H. | title = Measure Theory | volume = 2 | year = 2010 | edition = Second | page = 221}}</ref> This property is used in connection with [[Lebesgue integral]].
{{math proof| proof=
Both <math>F(t) := \mu\{x\in X : f(x) > t\}</math> and <math>G(t) := \mu\{x\in X : f(x) \geq t\}</math> are monotonically non-increasing functions of <math>t,</math> so both of them have [[Discontinuities of monotone functions|at most countably many discontinuities]] and thus they are continuous almost everywhere, relative to the Lebesgue measure. 
If <math>t < 0</math> then <math>\{x\in X : f(x) \geq t\} = X = \{x\in X : f(x) > t\},</math> so that <math>F(t) = G(t),</math> as desired. 

If <math>t</math> is such that <math>\mu\{x\in X : f(x) > t\} = +\infty</math> then [[#Monotonicity|monotonicity]] implies
<math display=block>\mu\{x\in X : f(x) \geq t\} = +\infty,</math>
so that <math>F(t) = G(t),</math> as required. 
If <math>\mu\{x\in X : f(x) > t\} = +\infty</math> for all <math>t</math> then we are done, so assume otherwise. Then there is a unique <math>t_0 \in \{-\infty\} \cup [0,+\infty) </math> such that <math>F</math> is infinite to the left of <math>t</math> (which can only happen when <math>t_0 \geq 0</math>) and finite to the right. 
Arguing as above, <math>\mu\{x\in X : f(x) \geq t\} = +\infty </math> when <math>t < t_0.</math> Similarly, if <math>t_0 \geq 0</math> and <math>F\left(t_0\right) = +\infty</math> then <math>F\left(t_0\right) = G\left(t_0\right).</math>

For <math>t > t_0,</math> let <math>t_n</math> be a monotonically non-decreasing sequence converging to <math>t.</math> The monotonically non-increasing sequences <math>\{x\in X : f(x) > t_n\}</math> of members of <math>\Sigma</math> has at least one finitely <math>\mu</math>-measurable component, and
<math display=block>\{x\in X : f(x) \geq t\} = \bigcap_n \{x\in X : f(x) > t_n\}.</math>
Continuity from above guarantees that
<math display=block>\mu\{x\in X : f(x) \geq t\} = \lim_{t_n \uparrow t} \mu\{x\in X : f(x) > t_n\}.</math>
The right-hand side <math>\lim_{t_n \uparrow t} F\left(t_n\right)</math> then equals <math>F(t) = \mu\{x\in X : f(x) > t\}</math> if <math>t</math> is a point of continuity of <math>F.</math> Since <math>F</math> is continuous almost everywhere, this completes the proof.
}}

===Additivity===
Measures are required to be countably additive. However, the condition can be strengthened as follows.
For any set <math>I</math> and any set of nonnegative <math>r_i,i\in I</math> define:
<math display=block>\sum_{i\in I} r_i=\sup\left\lbrace\sum_{i\in J} r_i : |J|<\infty, J\subseteq I\right\rbrace.</math>
That is, we define the sum of the <math>r_i</math> to be the supremum of all the sums of finitely many of them.

A measure <math>\mu</math> on <math>\Sigma</math> is <math>\kappa</math>-additive if for any <math>\lambda<\kappa</math> and any family of disjoint sets <math>X_\alpha,\alpha<\lambda</math> the following hold:
<math display=block>\bigcup_{\alpha\in\lambda} X_\alpha \in \Sigma</math>
<math display=block>\mu\left(\bigcup_{\alpha\in\lambda} X_\alpha\right) = \sum_{\alpha\in\lambda}\mu\left(X_\alpha\right).</math>
The second condition is equivalent to the statement that the [[Ideal (set theory)|ideal]] of null sets is <math>\kappa</math>-complete.

===Sigma-finite measures===
{{Main|Sigma-finite measure}}

A measure space <math>(X, \Sigma, \mu)</math> is called finite if <math>\mu(X)</math> is a finite real number (rather than <math>\infty</math>). Nonzero finite measures are analogous to [[probability measure]]s in the sense that any finite measure <math>\mu</math> is proportional to the probability measure <math>\frac{1}{\mu(X)}\mu.</math> A measure <math>\mu</math> is called ''σ-finite'' if <math>X</math> can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a ''σ-finite measure'' if it is a countable union of sets with finite measure.

For example, the [[real number]]s with the standard [[Lebesgue measure]] are σ-finite but not finite. Consider the [[closed interval]]s <math>[k, k+1]</math> for all [[integer]]s <math>k;</math> there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the [[real number]]s with the [[counting 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 the [[Lindelöf space|Lindelöf property]] of topological spaces.{{OR inline|date=May 2022}} They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.

===Strictly localizable measures===
{{Main|Decomposable measure}}

===Semifinite measures===
Let <math>X</math> be a set, let <math>{\cal A}</math> be a sigma-algebra on <math>X,</math> and let <math>\mu</math> be a measure on <math>{\cal A}.</math> We say <math>\mu</math> is '''semifinite''' to mean that for all <math>A\in\mu^\text{pre}\{+\infty\},</math> <math>{\cal P}(A)\cap\mu^\text{pre}(\R_{>0})\ne\emptyset.</math>{{sfn|Mukherjea|Pothoven|1985|p=90}} 

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====
* Every sigma-finite measure is semifinite.
* Assume <math>{\cal A}={\cal P}(X),</math> let <math>f:X\to[0,+\infty],</math> and assume <math>\mu(A)=\sum_{a\in A}f(a)</math> for all <math>A\subseteq X.</math>
** We have that <math>\mu</math> is sigma-finite if and only if <math>f(x)<+\infty</math> for all <math>x\in X</math> and <math>f^\text{pre}(\R_{>0})</math> is countable. We have that <math>\mu</math> is semifinite if and only if <math>f(x)<+\infty</math> for all <math>x\in X.</math>{{sfn|Folland|1999|p=25}}
** Taking <math>f=X\times\{1\}</math> above (so that <math>\mu</math> is counting measure on <math>{\cal P}(X)</math>), we see that counting measure on <math>{\cal P}(X)</math> is
*** sigma-finite if and only if <math>X</math> is countable; and
*** semifinite (without regard to whether <math>X</math> is countable). (Thus, counting measure, on the power set <math>{\cal P}(X)</math> of an arbitrary uncountable set <math>X,</math> gives an example of a semifinite measure that is not sigma-finite.)
* Let <math>d</math> be a complete, separable metric on <math>X,</math> let <math>{\cal B}</math> be the [[Borel sigma-algebra]] induced by <math>d,</math> and let <math>s\in\R_{>0}.</math> Then the [[Hausdorff measure]] <math>{\cal H}^s|{\cal B}</math> is semifinite.{{sfn|Edgar|1998|loc=Theorem 1.5.2, p. 42}}
* Let <math>d</math> be a complete, separable metric on <math>X,</math> let <math>{\cal B}</math> be the [[Borel sigma-algebra]] induced by <math>d,</math> and let <math>s\in\R_{>0}.</math> Then the [[Packing dimension#Definitions|packing measure]] <math>{\cal H}^s|{\cal B}</math> is semifinite.{{sfn|Edgar|1998|loc=Theorem 1.5.3, p. 42}}

====Involved example====
The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal to <math>\mu.</math> It can be shown there is a greatest measure with these two properties:
{{Math theorem|name=Theorem (semifinite part){{sfn|Nielsen|1997|loc=Exercise 11.30, p. 159}}|math_statement=
For any measure <math>\mu</math> on <math>{\cal A},</math> there exists, among semifinite measures on <math>{\cal A}</math> that are less than or equal to <math>\mu,</math> a [[Greatest element and least element|greatest]] element <math>\mu_\text{sf}.</math>
}}
We say the '''semifinite part''' of <math>\mu</math> to mean the semifinite measure <math>\mu_\text{sf}</math> defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for the semifinite part:
* <math>\mu_\text{sf}=(\sup\{\mu(B):B\in{\cal P}(A)\cap\mu^\text{pre}(\R_{\ge0})\})_{A\in{\cal A}}.</math>{{sfn|Nielsen|1997|loc=Exercise 11.30, p. 159}}
* <math>\mu_\text{sf}=(\sup\{\mu(A\cap B):B\in\mu^\text{pre}(\R_{\ge0})\})_{A\in{\cal A}}\}.</math>{{sfn|Fremlin|2016|loc=Section 213X, part (c)}}
* <math>\mu_\text{sf}=\mu|_{\mu^\text{pre}(\R_{>0})}\cup\{A\in{\cal A}:\sup\{\mu(B):B\in{\cal P}(A)\}=+\infty\}\times\{+\infty\}\cup\{A\in{\cal A}:\sup\{\mu(B):B\in{\cal P}(A)\}<+\infty\}\times\{0\}.</math>{{sfn|Royden|Fitzpatrick|2010|loc=Exercise 17.8, p. 342}}

Since <math>\mu_\text{sf}</math> is semifinite, it follows that if <math>\mu=\mu_\text{sf}</math> then <math>\mu</math> is semifinite. It is also evident that if <math>\mu</math> is semifinite then <math>\mu=\mu_\text{sf}.</math>

====Non-examples====
Every ''<math>0-\infty</math> measure'' that is not the zero measure is not semifinite. (Here, we say ''<math>0-\infty</math> measure'' to mean a measure whose range lies in <math>\{0,+\infty\}</math>: <math>(\forall A\in{\cal A})(\mu(A)\in\{0,+\infty\}).</math>) Below we give examples of <math>0-\infty</math> measures that are not zero measures.
* Let <math>X</math> be nonempty, let <math>{\cal A}</math> be a <math>\sigma</math>-algebra on <math>X,</math> let <math>f:X\to\{0,+\infty\}</math> be not the zero function, and let <math>\mu=(\sum_{x\in A}f(x))_{A\in{\cal A}}.</math> It can be shown that <math>\mu</math> is a measure.
** <math>\mu=\{(\emptyset,0)\}\cup({\cal A}\setminus\{\emptyset\})\times\{+\infty\}.</math>{{sfn|Hewitt|Stromberg|1965|loc=part (b) of Example 10.4, p. 127}}
*** <math>X=\{0\},</math> <math>{\cal A}=\{\emptyset,X\},</math> <math>\mu=\{(\emptyset,0),(X,+\infty)\}.</math>{{sfn|Fremlin|2016|loc=Section 211O, p. 15}}
* Let <math>X</math> be uncountable, let <math>{\cal A}</math> be a <math>\sigma</math>-algebra on <math>X,</math> let <math>{\cal C}=\{A\in{\cal A}:A\text{ is countable}\}</math> be the countable elements of <math>{\cal A},</math> and let <math>\mu={\cal C}\times\{0\}\cup({\cal A}\setminus{\cal C})\times\{+\infty\}.</math> It can be shown that <math>\mu</math> is a measure.{{sfn|Mukherjea|Pothoven|1985|p=90}}

====Involved non-example====
{{Blockquote
|text=Measures that are not semifinite are very wild when restricted to certain sets.<ref group=Note>One way to rephrase our definition is that <math>\mu</math> is semifinite if and only if <math>(\forall A\in\mu^\text{pre}\{+\infty\})(\exists B\subseteq A)(0<\mu(B)<+\infty).</math> Negating this rephrasing, we find that <math>\mu</math> is not semifinite if and only if <math>(\exists A\in\mu^\text{pre}\{+\infty\})(\forall B\subseteq A)(\mu(B)\in\{0,+\infty\}).</math> For every such set <math>A,</math> the subspace measure induced by the subspace sigma-algebra induced by <math>A,</math> i.e. the restriction of <math>\mu</math> to said subspace sigma-algebra, is a <math>0-\infty</math> measure that is not the zero measure.</ref> Every measure is, in a sense, semifinite once its <math>0-\infty</math> part (the wild part) is taken away.
|author=A. Mukherjea and K. Pothoven
|source=''Real and Functional Analysis, Part A: Real Analysis'' (1985)
}}
{{Math theorem|name=Theorem (Luther decomposition){{sfn|Luther|1967|loc=Theorem 1}}{{sfn|Mukherjea|Pothoven|1985|loc=part (b) of Proposition 2.3, p. 90}}|math_statement=
For any measure <math>\mu</math> on <math>{\cal A},</math> there exists a <math>0-\infty</math> measure <math>\xi</math> on <math>{\cal A}</math> such that <math>\mu=\nu+\xi</math> for some semifinite measure <math>\nu</math> on <math>{\cal A}.</math> In fact, among such measures <math>\xi,</math> there exists a [[Greatest element and least element|least]] measure <math>\mu_{0-\infty}.</math> Also, we have <math>\mu=\mu_\text{sf}+\mu_{0-\infty}.</math>
}}
We say the '''<math>\mathbf{0-\infty}</math> part''' of <math>\mu</math> to mean the measure <math>\mu_{0-\infty}</math> defined in the above theorem. Here is an explicit formula for <math>\mu_{0-\infty}</math>: <math>\mu_{0-\infty}=(\sup\{\mu(B)-\mu_\text{sf}(B):B\in{\cal P}(A)\cap\mu_\text{sf}^\text{pre}(\R_{\ge0})\})_{A\in{\cal A}}.</math>

====Results regarding semifinite measures====
<!--Help appreciated, please add results / check the c.l.d. product measure thing-->
* Let <math>\mathbb F</math> be <math>\R</math> or <math>\C,</math> and let <math>T:L_\mathbb{F}^\infty(\mu)\to\left(L_\mathbb{F}^1(\mu)\right)^*:g\mapsto T_g=\left(\int fgd\mu\right)_{f\in L_\mathbb{F}^1(\mu)}.</math> Then <math>\mu</math> is semifinite if and only if <math>T</math> is injective.{{sfn|Fremlin|2016|loc=part (a) of Theorem 243G, p. 159}}{{sfn|Fremlin|2016|loc=Section 243K, p. 162}} (This result has import in the study of the [[Lp space#Dual spaces|dual space of <math>L^1=L_\mathbb{F}^1(\mu)</math>]].)
* Let <math>\mathbb F</math> be <math>\R</math> or <math>\C,</math> and let <math>{\cal T}</math> be the topology of convergence in measure on <math>L_\mathbb{F}^0(\mu).</math> Then <math>\mu</math> is semifinite if and only if <math>{\cal T}</math> is Hausdorff.{{sfn|Fremlin|2016|loc=part (a) of the Theorem in Section 245E, p. 182}}{{sfn|Fremlin|2016|loc=Section 245M, p. 188}}
* (Johnson) Let <math>X</math> be a set, let <math>{\cal A}</math> be a sigma-algebra on <math>X,</math> let <math>\mu</math> be a measure on <math>{\cal A},</math> let <math>Y</math> be a set, let <math>{\cal B}</math> be a sigma-algebra on <math>Y,</math> and let <math>\nu</math> be a measure on <math>{\cal B}.</math> If <math>\mu,\nu</math> are both not a <math>0-\infty</math> measure, then both <math>\mu</math> and <math>\nu</math> are semifinite if and only if [[Product measure|<math>(\mu\times_\text{cld}\nu)</math>]]<math>(A\times B)=\mu(A)\nu(B)</math> for all <math>A\in{\cal A}</math> and <math>B\in{\cal B}.</math> (Here, <math>\mu\times_\text{cld}\nu</math> is the measure defined in Theorem 39.1 in Berberian '65.{{sfn|Berberian|1965|loc=Theorem 39.1, p. 129}}) <!--To check: Is this actually the ''c.l.d. product measure''?{{sfn|Fremlin|2016|loc=Definition 251F, p. 206}}-->

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

Let <math>X</math> be a set, let <math>{\cal A}</math> be a sigma-algebra on <math>X,</math> and let <math>\mu</math> be a measure on <math>{\cal A}.</math>
* Let <math>\mathbb F</math> be <math>\R</math> or <math>\C,</math> and let <math>T : L_\mathbb{F}^\infty(\mu) \to \left(L_\mathbb{F}^1(\mu)\right)^* : g \mapsto T_g = \left(\int fgd\mu\right)_{f\in L_\mathbb{F}^1(\mu)}.</math> Then <math>\mu</math> is localizable if and only if <math>T</math> is bijective (if and only if <math>L_\mathbb{F}^\infty(\mu)</math> "is" <math>L_\mathbb{F}^1(\mu)^*</math>).{{sfn|Fremlin|2016|loc=part (b) of Theorem 243G, p. 159}}{{sfn|Fremlin|2016|loc=Section 243K, p. 162}}

===s-finite measures===
{{Main|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 of [[stochastic processes]].

==Non-measurable sets==
{{Main|Non-measurable set}}

If the [[axiom of choice]] is assumed to be true, it can be proved that not all subsets of [[Euclidean space]] are [[Lebesgue measurable]]; examples of such sets include the [[Vitali set]], and the non-measurable sets postulated by the [[Hausdorff paradox]] and the [[Banach–Tarski paradox]].

==Generalizations==
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 additive [[set function]] with values in the (signed) real numbers is called a ''[[signed measure]]'', while such a function with values in the [[complex numbers]] is called a ''[[complex measure]]''. Observe, however, that complex measure is necessarily of finite [[total variation|variation]], hence complex measures include [[Finite measure|finite signed measures]] but not, for example, the [[Lebesgue measure]].

Measures that take values in [[Banach spaces]] have been studied extensively.<ref>{{citation
 | last = Rao | first = M. M.
 | isbn = 978-981-4350-81-5
 | mr = 2840012
 | publisher = [[World Scientific]]
 | series = Series on Multivariate Analysis
 | title = Random and Vector Measures
 | volume = 9
 | year = 2012}}.</ref> A measure that takes values in the set of self-adjoint projections on a [[Hilbert space]] is called a ''[[projection-valued measure]]''; these are used in [[functional analysis]] for the [[spectral theorem]]. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term '''positive measure''' is used. Positive measures are closed under [[conical combination]] but not general [[linear combination]], while signed measures are the linear closure of positive measures. More generally see [[measure theory in topological vector spaces]].

Another generalization is the ''finitely additive measure'', also known as a [[Content (measure theory)|content]]. This is the same as a measure except that instead of requiring ''countable'' additivity we require only ''finite'' additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected with notions such as [[Banach limit]]s, the dual of [[lp space|<math>L^\infty</math>]] and the [[Stone–Čech compactification]]. All these are linked in one way or another to the [[axiom of choice]].  Contents remain useful in certain technical problems in [[geometric measure theory]]; this is the theory of [[Banach measure]]s.

A ''charge'' is a generalization in both directions: it is a finitely additive, signed measure.<ref>{{Cite book|last=Bhaskara Rao|first=K. P. S.|url=https://www.worldcat.org/oclc/21196971|title=Theory of charges: a study of finitely additive measures|date=1983|publisher=Academic Press|others=M. Bhaskara Rao|isbn=0-12-095780-9|location=London|pages=35|oclc=21196971}}</ref> (Cf. [[ba space]] for information about ''bounded'' charges, where we say a charge is ''bounded'' to mean its range its a bounded subset of ''R''.)

==See also==
{{Portal|Mathematics}}
{{div col|colwidth=20em|small=yes}}
* [[Abelian von Neumann algebra]]
* [[Almost everywhere]]
* [[Carathéodory's extension theorem]]
* [[Content (measure theory)]]
* [[Fubini's theorem]]
* [[Fatou's lemma]]
* [[Fuzzy measure theory]]
* [[Geometric measure theory]]
* [[Hausdorff measure]]
* [[Inner measure]]
* [[Lebesgue integration]]
* [[Lebesgue measure]]
* [[Lorentz space]]
* [[Lifting theory]]
* [[Measurable cardinal]]
* [[Measurable function]]
* [[Minkowski content]]
* [[Outer measure]]
* [[Product measure]]
* [[Pushforward measure]]
* [[Regular measure]]
* [[Vector measure]]
* [[Valuation (measure theory)]]
* [[Volume form]]
{{div col end}}

==Notes==
{{reflist|group=Note}}

==Bibliography==
{{refbegin}}
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* {{citation | last=Bauer|first=Heinz|authorlink=Heinz Bauer|title=Measure and Integration Theory|year=2001|publisher=de Gruyter|location=Berlin|isbn=978-3110167191}}
* {{citation | last=Bear|first=H.S.|title=A Primer of Lebesgue Integration|year=2001|publisher=Academic Press|location=San Diego|isbn=978-0120839711}}
* {{cite book |last1=Berberian |first1=Sterling K |title=Measure and Integration |date=1965 |publisher=MacMillan}}
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* {{cite book |last=Dudley|first=Richard M.|authorlink=Richard M. Dudley| title=Real Analysis and Probability | year=2002 |publisher=Cambridge University Press|isbn=978-0521007542}}
* {{cite book |last1=Edgar |first1=Gerald A. |title=Integral, Probability, and Fractal Measures |date=1998 |publisher=Springer |isbn=978-1-4419-3112-2}}
* {{cite book |last1=Folland |first1=Gerald B. |authorlink=Gerald Folland|title=Real Analysis: Modern Techniques and Their Applications |date=1999 |publisher=Wiley |isbn=0-471-31716-0 |edition=Second}}
* Federer, Herbert. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp.
* {{cite book |last1=Fremlin |first1=D.H. |title=Measure Theory, Volume 2: Broad Foundations |date=2016 |publisher=Torres Fremlin |edition=Hardback}} Second printing.
* {{cite book |last1=Hewitt |first1=Edward |last2=Stromberg |first2=Karl |title=Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable |date=1965 |publisher=Springer |isbn=0-387-90138-8}}
* {{citation | last=Jech| first=Thomas | title=Set Theory: The Third Millennium Edition, Revised and Expanded |  year=2003 | publisher=[[Springer Verlag]] | isbn=3-540-44085-2}}
* [[R. Duncan Luce]] and Louis Narens (1987). "measurement, theory of", ''The [[New Palgrave: A Dictionary of Economics]]'', v. 3, pp.&nbsp;428–32.
* {{cite journal |last1=Luther |first1=Norman Y |title=A decomposition of measures |journal=Canadian Journal of Mathematics |date=1967 |volume=20 |pages=953–959 |doi=10.4153/CJM-1968-092-0 |s2cid=124262782 |doi-access=free }}
* {{cite book |last1=Mukherjea |first1=A |last2=Pothoven |first2=K |title=Real and Functional Analysis, Part A: Real Analysis |date=1985 |publisher=Plenum Press |edition=Second |url=https://link.springer.com/book/9781441974587}}
** The first edition was published with ''Part B: Functional Analysis'' as a single volume: {{cite book |last1=Mukherjea |first1=A |last2=Pothoven |first2=K |title=Real and Functional Analysis |date=1978 |publisher=Plenum Press |doi=10.1007/978-1-4684-2331-0 |isbn=978-1-4684-2333-4 |edition=First |url=https://link.springer.com/book/10.1007/978-1-4684-2331-0}}
* M. E. Munroe, 1953. ''Introduction to Measure and Integration''. Addison Wesley.
* {{cite book |last1=Nielsen |first1=Ole A |title=An Introduction to Integration and Measure Theory |date=1997 |publisher=Wiley |isbn=0-471-59518-7}}
* {{Citation|author=K. P. S. Bhaskara Rao and M. Bhaskara Rao|title=Theory of Charges: A Study of Finitely Additive Measures| publisher=Academic Press|location=London|year=1983|pages=x + 315|isbn=0-12-095780-9}}
* {{cite book |last1=Royden |first1=H.L. |last2=Fitzpatrick |first2=P.M. |author1-link=Halsey Royden |title=Real Analysis |date=2010 |publisher=Prentice Hall |page=342, Exercise 17.8 |edition=Fourth}} First printing. There is a later (2017) second printing. Though usually there is little difference between the first and subsequent printings, in this case the second printing not only deletes from page 53 the Exercises 36, 40, 41, and 42 of Chapter 2 but also offers a (slightly, but still substantially) different presentation of part (ii) of Exercise 17.8. (The second printing's presentation of part (ii) of Exercise 17.8 (on the Luther{{sfn|Luther|1967|loc=Theorem 1}} decomposition) agrees with usual presentations,{{sfn|Mukherjea|Pothoven|1985|p=90}}{{sfn|Folland|1999|p=27|loc=Exercise 1.15.a}} whereas the first printing's presentation provides a fresh perspective.)
* Shilov, G. E., and Gurevich, B. L., 1978. ''Integral, Measure, and Derivative: A Unified Approach'', Richard A. Silverman, trans. Dover Publications. {{isbn|0-486-63519-8}}. Emphasizes the [[Daniell integral]].
* {{citation | last = Teschl| first = Gerald| author-link = Gerald Teschl| title = Topics in Real Analysis| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ra/index.html|publisher = (lecture notes)}}
* {{cite book|last1=Tao|first1=Terence|author-link=Terence Tao|title=An Introduction to Measure Theory|date=2011|publisher=American Mathematical Society|location=Providence, R.I.|isbn=9780821869192}}
* {{cite book|last1=Weaver|first1=Nik|title=Measure Theory and Functional Analysis|date=2013|publisher= [[World Scientific]]|isbn=9789814508568}}
{{refend}}

==References==
{{reflist}}

==External links==
{{Wiktionary|measurable}}
* {{springer|title=Measure|id=p/m063240}}
* [https://vannevar.ece.uw.edu/techsite/papers/documents/UWEETR-2006-0008.pdf Tutorial: Measure Theory for Dummies]

{{Measure theory}}
{{Lp spaces}}

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[[Category:Measure theory| ]]
[[Category:Measures (measure theory)| ]]