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, 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, integration theory, and can be generalized to assume negative values, as with electrical charge. Farreaching generalizations (such as spectral measures and projectionvalued measures) 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. 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.
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Transcription
Definition
Let be a set and a algebra over A set function from to the extended real number line is called a measure if the following conditions hold:
 Nonnegativity: For all
 Countable additivity (or additivity): For all countable collections of pairwise disjoint sets in Σ,
If at least one set has finite measure, then the requirement is met automatically due to countable additivity:
If the condition of nonnegativity is dropped, and takes on at most one of the values of then is called a signed measure.
The pair is called a measurable space, and the members of are called measurable sets.
A triple is called a measure space. A probability measure is a measure with total measure one – that is, A probability space is a measure space with a probability measure.
For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex topological vector space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures.
Instances
Some important measures are listed here.
 The counting measure is defined by = number of elements in
 The Lebesgue measure on is a complete translationinvariant measure on a σalgebra containing the intervals in such that ; and every other measure with these properties extends 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 topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.
 The Hausdorff measure is a generalization of the Lebesgue measure to sets with noninteger 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_{a} (cf. Dirac delta function) is given by δ_{a}(S) = χ_{S}(a), where χ_{S} is the indicator function of The measure of a set is 1 if it contains the point 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 nonnegative extensive property, conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.
 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.
Basic properties
Let be a measure.
Monotonicity
If and are measurable sets with then
Measure of countable unions and intersections
Countable subadditivity
For any countable sequence of (not necessarily disjoint) measurable sets in
Continuity from below
If are measurable sets that are increasing (meaning that ) then the union of the sets is measurable and
Continuity from above
If are measurable sets that are decreasing (meaning that ) then the intersection of the sets is measurable; furthermore, if at least one of the has finite measure then
This property is false without the assumption that at least one of the has finite measure. For instance, for each let which all have infinite Lebesgue measure, but the intersection is empty.
Other properties
Completeness
A measurable set is called a null set if 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 σalgebra of subsets which differ by a negligible set from a measurable set that is, such that the symmetric difference of and is contained in a null set. One defines to equal
μ{x : f(x) ≥ t} = μ{x : f(x) > t} (a.e.)
If is measurable, then
Both and are monotonically nonincreasing functions of so both of them have at most countably many discontinuities and thus they are continuous almost everywhere, relative to the Lebesgue measure. If then so that as desired.
If is such that then monotonicity implies
For let be a monotonically nondecreasing sequence converging to The monotonically nonincreasing sequence of members of has at least one finitely measurable component, and
Additivity
Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set and any set of nonnegative define:
A measure on is additive if for any and any family of disjoint sets the following hold:
Sigmafinite measures
A measure space is called finite if is a finite real number (rather than ). Nonzero finite measures are analogous to probability measures in the sense that any finite measure is proportional to the probability measure A measure is called σfinite if 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 σfinite measure if it is a countable union of sets with finite measure.
For example, the real numbers with the standard Lebesgue measure are σfinite but not finite. Consider the closed intervals for all integers there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σfinite, 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 σfinite measure spaces have some very convenient properties; σfiniteness can be compared in this respect to the Lindelöf 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
Semifinite measures
Let be a set, let be a sigmaalgebra on and let be a measure on We say is semifinite to mean that for all ^{[2]}
Semifinite measures generalize sigmafinite measures, in such a way that some big theorems of measure theory that hold for sigmafinite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (Todo: add examples of such theorems; cf. the talk page.)
Basic examples
 Every sigmafinite measure is semifinite.
 Assume let and assume for all
 We have that is sigmafinite if and only if for all and is countable. We have that is semifinite if and only if for all ^{[3]}
 Taking above (so that is counting measure on ), we see that counting measure on is
 sigmafinite if and only if is countable; and
 semifinite (without regard to whether is countable). (Thus, counting measure, on the power set of an arbitrary uncountable set gives an example of a semifinite measure that is not sigmafinite.)
 Let be a complete, separable metric on let be the Borel sigmaalgebra induced by and let Then the Hausdorff measure is semifinite.^{[4]}
 Let be a complete, separable metric on let be the Borel sigmaalgebra induced by and let Then the packing measure is semifinite.^{[5]}
Involved example
The zero measure is sigmafinite and thus semifinite. In addition, the zero measure is clearly less than or equal to It can be shown there is a greatest measure with these two properties:
Theorem (semifinite part)^{[6]} — For any measure on there exists, among semifinite measures on that are less than or equal to a greatest element
We say the semifinite part of to mean the semifinite measure defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for the semifinite part:
 ^{[6]}
 ^{[7]}
 ^{[8]}
Since is semifinite, it follows that if then is semifinite. It is also evident that if is semifinite then
Nonexamples
Every measure that is not the zero measure is not semifinite. (Here, we say measure to mean a measure whose range lies in : ) Below we give examples of measures that are not zero measures.
 Let be nonempty, let be a algebra on let be not the zero function, and let It can be shown that is a measure.
 ^{[9]}
 ^{[10]}
 ^{[9]}
 Let be uncountable, let be a algebra on let be the countable elements of and let It can be shown that is a measure.^{[2]}
Involved nonexample
Measures that are not semifinite are very wild when restricted to certain sets.^{[Note 1]} Every measure is, in a sense, semifinite once its part (the wild part) is taken away.
— A. Mukherjea and K. Pothoven, Real and Functional Analysis, Part A: Real Analysis (1985)
Theorem (Luther decomposition)^{[11]}^{[12]} — For any measure on there exists a measure on such that for some semifinite measure on In fact, among such measures there exists a least measure Also, we have
We say the part of to mean the measure defined in the above theorem. Here is an explicit formula for :
Results regarding semifinite measures
 Let be or and let Then is semifinite if and only if is injective.^{[13]}^{[14]} (This result has import in the study of the dual space of .)
 Let be or and let be the topology of convergence in measure on Then is semifinite if and only if is Hausdorff.^{[15]}^{[16]}
 (Johnson) Let be a set, let be a sigmaalgebra on let be a measure on let be a set, let be a sigmaalgebra on and let be a measure on If are both not a measure, then both and are semifinite if and only if for all and (Here, is the measure defined in Theorem 39.1 in Berberian '65.^{[17]})
Localizable measures
Localizable measures are a special case of semifinite measures and a generalization of sigmafinite measures.
Let be a set, let be a sigmaalgebra on and let be a measure on
 Let be or and let Then is localizable if and only if is bijective (if and only if "is" ).^{[18]}^{[14]}
sfinite measures
A measure is said to be sfinite if it is a countable sum of finite measures. Sfinite measures are more general than sigmafinite ones and have applications in the theory of stochastic processes.
Nonmeasurable sets
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 nonmeasurable 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 nonnegative 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 variation, hence complex measures include finite signed measures but not, for example, the Lebesgue measure.
Measures that take values in Banach spaces have been studied extensively.^{[19]} A measure that takes values in the set of selfadjoint projections on a Hilbert space is called a projectionvalued measure; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take nonnegative 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.
Another generalization is the finitely additive measure, also known as a 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 limits, the dual of 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 measures.
A charge is a generalization in both directions: it is a finitely additive, signed measure.^{[20]} (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
 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
Notes
 ^ One way to rephrase our definition is that is semifinite if and only if Negating this rephrasing, we find that is not semifinite if and only if For every such set the subspace measure induced by the subspace sigmaalgebra induced by i.e. the restriction of to said subspace sigmaalgebra, is a measure that is not the zero measure.
Bibliography
 Robert G. Bartle (1995) The Elements of Integration and Lebesgue Measure, Wiley Interscience.
 Bauer, H. (2001), Measure and Integration Theory, Berlin: de Gruyter, ISBN 9783110167191
 Bear, H.S. (2001), A Primer of Lebesgue Integration, San Diego: Academic Press, ISBN 9780120839711
 Berberian, Sterling K (1965). Measure and Integration. MacMillan.
 Bogachev, V. I. (2006), Measure theory, Berlin: Springer, ISBN 9783540345138
 Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3540411291 Chapter III.
 R. M. Dudley, 2002. Real Analysis and Probability. Cambridge University Press.
 Edgar, Gerald A (1998). Integral, Probability, and Fractal Measures. Springer. ISBN 9781441931122.
 Folland, Gerald B (1999). Real Analysis: Modern Techniques and Their Applications (Second ed.). Wiley. ISBN 0471317160.
 Federer, Herbert. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 SpringerVerlag New York Inc., New York 1969 xiv+676 pp.
 Fremlin, D.H. (2016). Measure Theory, Volume 2: Broad Foundations (Hardback ed.). Torres Fremlin. Second printing.
 Hewitt, Edward; Stromberg, Karl (1965). Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable. Springer. ISBN 0387901388.
 Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Revised and Expanded, Springer Verlag, ISBN 3540440852
 R. Duncan Luce and Louis Narens (1987). "measurement, theory of", The New Palgrave: A Dictionary of Economics, v. 3, pp. 428–32.
 Luther, Norman Y (1967). "A decomposition of measures". Canadian Journal of Mathematics. 20: 953–959. doi:10.4153/CJM19680920. S2CID 124262782.
 Mukherjea, A; Pothoven, K (1985). Real and Functional Analysis, Part A: Real Analysis (Second ed.). Plenum Press.
 The first edition was published with Part B: Functional Analysis as a single volume: Mukherjea, A; Pothoven, K (1978). Real and Functional Analysis (First ed.). Plenum Press. doi:10.1007/9781468423310. ISBN 9781468423334.
 M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley.
 Nielsen, Ole A (1997). An Introduction to Integration and Measure Theory. Wiley. ISBN 0471595187.
 K. P. S. Bhaskara Rao and M. Bhaskara Rao (1983), Theory of Charges: A Study of Finitely Additive Measures, London: Academic Press, pp. x + 315, ISBN 0120957809
 Royden, H.L.; Fitzpatrick, P.M. (2010). Real Analysis (Fourth ed.). Prentice Hall. p. 342, Exercise 17.8. 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^{[11]} decomposition) agrees with usual presentations,^{[2]}^{[21]} 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 0486635198. Emphasizes the Daniell integral.
 Teschl, Gerald, Topics in Real and Functional Analysis, (lecture notes)
 Tao, Terence (2011). An Introduction to Measure Theory. Providence, R.I.: American Mathematical Society. ISBN 9780821869192.
 Weaver, Nik (2013). Measure Theory and Functional Analysis. World Scientific. ISBN 9789814508568.
References
 ^ Fremlin, D. H. (2010), Measure Theory, vol. 2 (Second ed.), p. 221
 ^ ^{a} ^{b} ^{c} Mukherjea 1985, p. 90.
 ^ Folland 1999, p. 25.
 ^ Edgar 1998, Theorem 1.5.2, p. 42.
 ^ Edgar 1998, Theorem 1.5.3, p. 42.
 ^ ^{a} ^{b} Nielsen 1997, Exercise 11.30, p. 159.
 ^ Fremlin 2016, Section 213X, part (c).
 ^ Royden 2010, Exercise 17.8, p. 342.
 ^ Hewitt 1965, part (b) of Example 10.4, p. 127.
 ^ Fremlin 2016, Section 211O, p. 15.
 ^ ^{a} ^{b} Luther 1967, Theorem 1.
 ^ Mukherjea 1985, part (b) of Proposition 2.3, p. 90.
 ^ Fremlin 2016, part (a) of Theorem 243G, p. 159.
 ^ ^{a} ^{b} Fremlin 2016, Section 243K, p. 162.
 ^ Fremlin 2016, part (a) of the Theorem in Section 245E, p. 182.
 ^ Fremlin 2016, Section 245M, p. 188.
 ^ Berberian 1965, Theorem 39.1, p. 129.
 ^ Fremlin 2016, part (b) of Theorem 243G, p. 159.
 ^ Rao, M. M. (2012), Random and Vector Measures, Series on Multivariate Analysis, vol. 9, World Scientific, ISBN 9789814350815, MR 2840012.
 ^ Bhaskara Rao, K. P. S. (1983). Theory of charges: a study of finitely additive measures. M. Bhaskara Rao. London: Academic Press. p. 35. ISBN 0120957809. OCLC 21196971.
 ^ Folland 1999, p. 27, Exercise 1.15.a.