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A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space.

A measurable space consists of the first two components without a specific measure.

Definition

A measure space is a triple ${\displaystyle (X,{\mathcal {A}},\mu ),}$ where

• ${\displaystyle X}$ is a set
• ${\displaystyle {\mathcal {A}}}$ is a σ-algebra on the set ${\displaystyle X}$
• ${\displaystyle \mu }$ is a measure on ${\displaystyle (X,{\mathcal {A}})}$

In other words, a measure space consists of a measurable space ${\displaystyle (X,{\mathcal {A}})}$ together with a measure on it.

Example

Set ${\displaystyle X=\{0,1\}}$. The ${\textstyle \sigma }$-algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by ${\textstyle \wp (\cdot ).}$ Sticking with this convention, we set $${\displaystyle {\mathcal {A}}=\wp (X)}$$

In this simple case, the power set can be written down explicitly: $${\displaystyle \wp (X)=\{\varnothing ,\{0\},\{1\},\{0,1\}\}.}$$

As the measure, define ${\textstyle \mu }$ by $${\displaystyle \mu (\{0\})=\mu (\{1\})={\frac {1}{2}},}$$ so ${\textstyle \mu (X)=1}$ (by additivity of measures) and ${\textstyle \mu (\varnothing )=0}$ (by definition of measures).

This leads to the measure space ${\textstyle (X,\wp (X),\mu ).}$ It is a probability space, since ${\textstyle \mu (X)=1.}$ The measure ${\textstyle \mu }$ corresponds to the Bernoulli distribution with ${\textstyle p={\frac {1}{2}},}$ which is for example used to model a fair coin flip.

Important classes of measure spaces

Most important classes of measure spaces are defined by the properties of their associated measures. This includes, in order of increasing generality:

• Probability spaces, a measure space where the measure is a probability measure
• Finite measure spaces, where the measure is a finite measure
• ${\displaystyle \sigma }$-finite measure spaces, where the measure is a ${\displaystyle \sigma }$-finite measure

Another class of measure spaces are the complete measure spaces.

References

1. ^ Kosorok, Michael R. (2008). Introduction to Empirical Processes and Semiparametric Inference. New York: Springer. p. 83. ISBN 978-0-387-74977-8.
2. Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 18. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
3. ^ Anosov, D.V. (2001) , "Measure space", Encyclopedia of Mathematics, EMS Press
4. Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 33. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.

v t e Lp spaces
Basic concepts	Banach & Hilbert spaces L spaces Measure Lebesgue Measure space Measurable space/function Minkowski distance Sequence spaces
L spaces	Integrable function Lebesgue integration Taxicab geometry
L spaces	Bessel's Cauchy–Schwarz Euclidean distance Hilbert space Parseval's identity Polarization identity Pythagorean theorem Square-integrable function
${\displaystyle L^{\infty }}$ spaces	Bounded function Chebyshev distance Infimum and supremum Essential Uniform norm
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Results	Marcinkiewicz interpolation theorem Plancherel theorem Riemann–Lebesgue Riesz–Fischer theorem Riesz–Thorin theorem For Lebesgue measure Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality
Applications & related	Bochner space Fourier analysis Lorentz space Probability theory Quasinorm Real analysis Sobolev space *-algebra C*-algebra Von Neumann

• Measure theory
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