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Domain-specific¶
- Measurable space
- … underlying set, collection of subsets, empty-set inclusion, complement and countable-union closure, generated sigma-algebra if any, measurable maps and distinction from a measure space are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
This sourceIt captures and generalises intuitive notions such as length, area, and volume with a set X of 'points' in the space, but regions of the space are the elements of the σ-algebra, since the intuitive measures are not usually defined for points. The algebra also captures the relationships that might be expected of regions: that a region can be defined as an intersection of other regions, a union of other regions, or the space with the exception of another region. ==Definition== Consider a set X and a σ-algebra \mathcal F on X. Then the tuple (X, \mathcal F) is called a measurable space. The elements of \mathcal F are called 'measurable sets' within the measurable space. Note that in contrast to a measure space, no measure is needed for a measurable space. ==Example== Look at the set: X = \{1,2,3\}. One possible \sigma -algebra would be: \mathcal {F}_1 = \{X, \varnothing\}. Then \left(X, \mathcal{F}_1 \right) is a measurable space. Another possible \sigma -algebra would be the power set on X : \mathcal{F}_2 = \mathcal P(X). With this, a second measurable space on the set X is given by \left(X, \mathcal F_2\right). ==Common measurable spaces== If X is finite or countably infinite, the \sigma -algebra is most often the power set on X, so \mathcal{F} = \mathcal P(X). This leads to the measurable space (X, \mathcal P(X)). If X is a topological space, the \sigma -algebra is most commonly the Borel \sigma -algebra \mathcal B, so \mathcal{F} = \mathcal B(X). This leads to the measurable space (X, \mathcal B(X)) that is common for all topological spaces such as the real numbers \R. ==Ambiguity with Borel spaces== The term Borel space is used for different types of measurable spaces. It can refer to any measurable space, so it is a synonym for a measurable space as defined above a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel \sigma -algebra) ==See also== * Category of measurable spaces ==References==.
- … underlying set, collection of subsets, empty-set inclusion, complement and countable-union closure, generated sigma-algebra if any, measurable maps and distinction from a measure space are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
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