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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.
Given a (possibly incomplete) measure space (X, Σ, μ), there is an extension (X, Σ 0, μ 0) of this measure space that is complete. [3] The smallest such extension (i.e. the smallest σ-algebra Σ 0) is called the completion of the measure space. The completion can be constructed as follows:
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 [1] a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel -algebra) [3]
In mathematics — specifically, in measure theory — a perfect measure (or, more accurately, a perfect measure space) is one that is "well-behaved" in some sense. Intuitively, a perfect measure μ is one for which, if we consider the pushforward measure on the real line R, then every measurable set is "μ-approximately a Borel set".
There is an obvious algebraic duality between the vector space of all finitely additive measures σ on Σ and the vector space of simple functions (() = ()). It is easy to check that the linear form induced by σ is continuous in the sup-norm if σ is bounded, and the result follows since a linear form on the dense subspace of simple functions ...
This measure turns out to be the restriction to the Borel σ-algebra of the Lebesgue measure, which is a complete measure and is defined on the Lebesgue σ-algebra. The Lebesgue σ-algebra is actually the completion of the Borel σ-algebra, which means that it is the smallest σ-algebra that contains all the Borel sets and can be equipped with ...
Tools. Tools. move to sidebar ... Download as PDF; ... In mathematics, a Loeb space is a type of measure space introduced by Loeb using nonstandard ...
A metric space M is bounded if there is an r such that no pair of points in M is more than distance r apart. [b] The least such r is called the diameter of M. The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r. Every totally bounded space is bounded.