Measure space
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 [math]\displaystyle{ (X, \mathcal A, \mu), }[/math] where[1][2]
- [math]\displaystyle{ X }[/math] is a set
- [math]\displaystyle{ \mathcal A }[/math] is a σ-algebra on the set [math]\displaystyle{ X }[/math]
- [math]\displaystyle{ \mu }[/math] is a measure on [math]\displaystyle{ (X, \mathcal{A}) }[/math]
In other words, a measure space consists of a measurable space [math]\displaystyle{ (X, \mathcal{A}) }[/math] together with a measure on it.
Example
Set [math]\displaystyle{ X = \{0, 1\} }[/math]. The [math]\displaystyle{ \sigma }[/math]-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 [math]\displaystyle{ \wp(\cdot). }[/math] Sticking with this convention, we set [math]\displaystyle{ \mathcal{A} = \wp(X) }[/math]
In this simple case, the power set can be written down explicitly: [math]\displaystyle{ \wp(X) = \{\varnothing, \{0\}, \{1\}, \{0, 1\}\}. }[/math]
As the measure, define [math]\displaystyle{ \mu }[/math] by [math]\displaystyle{ \mu(\{0\}) = \mu(\{1\}) = \frac{1}{2}, }[/math] so [math]\displaystyle{ \mu(X) = 1 }[/math] (by additivity of measures) and [math]\displaystyle{ \mu(\varnothing) = 0 }[/math] (by definition of measures).
This leads to the measure space [math]\displaystyle{ (X, \wp(X), \mu). }[/math] It is a probability space, since [math]\displaystyle{ \mu(X) = 1. }[/math] The measure [math]\displaystyle{ \mu }[/math] corresponds to the Bernoulli distribution with [math]\displaystyle{ p = \frac{1}{2}, }[/math] 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[1]
- Finite measure spaces, where the measure is a finite measure[3]
- [math]\displaystyle{ \sigma }[/math]-finite measure spaces, where the measure is a [math]\displaystyle{ \sigma }[/math]-finite measure[3]
Another class of measure spaces are the complete measure spaces.[4]
References
- ↑ 1.0 1.1 Kosorok, Michael R. (2008). Introduction to Empirical Processes and Semiparametric Inference. New York: Springer. p. 83. ISBN 978-0-387-74977-8.
- ↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 18. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
- ↑ 3.0 3.1 Hazewinkel, Michiel, ed. (2001), "Measure space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Measure_space
- ↑ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 33. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
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