Measurable Set

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A Measurable Set is a set that is a member of a Measurable Space.



[math]\displaystyle{ \mu\left(\bigcup_{k=1}^\infty E_k\right)=\sum_{k=1}^\infty \mu(E_k) }[/math]
One may require that at least one set E has finite measure. Then the empty set automatically has measure zero because of countable additivity, because

[math]\displaystyle{ \mu(E)=\mu(E \cup \varnothing \cup \varnothing \cup \dots) = \mu(E) + \mu(\varnothing) + \mu(\varnothing) + \dots, }[/math]

which implies (since the sum on the right thus converges to a finite value) that [math]\displaystyle{ \mu(\varnothing)=0 }[/math].

If only the second and third conditions of the definition of measure above are met, and μ takes on at most one of the values ±∞, then μ is called a signed measure.

The pair (X, Σ) is called a measurable space, the members of Σ are called measurable sets. If [math]\displaystyle{ \left(X, \Sigma_X\right) }[/math] and [math]\displaystyle{ \left(Y, \Sigma_Y\right) }[/math] are two measurable spaces, then a function [math]\displaystyle{ f : X \to Y }[/math] is called measurable if for every Y-measurable set [math]\displaystyle{ B \in \Sigma_Y }[/math], the inverse image is X-measurable – i.e.: [math]\displaystyle{ f^{(-1)}(B) \in \Sigma_X }[/math]. In this setup, the composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows. See also Measurable function#Term usage variations about another setup.

A triple (X, Σ, μ) is called a measure space. A probability measure is a measure with total measure one – i.e. μ(X) = 1. 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 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.