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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 simple example is a volume (how big an object occupies a space) as a measure. In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and ...
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]
Formally, a metric measure space is a metric space equipped with a Borel regular measure such that every ball has positive measure. [21] For example Euclidean spaces of dimension n, and more generally n-dimensional Riemannian manifolds, naturally have the structure of a metric measure space, equipped with the Lebesgue measure.
A finite signed measure (a.k.a. real measure) is defined in the same way, except that it is only allowed to take real values. That is, it cannot take + or . Finite signed measures form a real vector space, while extended signed measures do not because they are not closed under addition. On the other hand, measures are extended signed measures ...
The measure f ∗ (λ) might also be called "arc length measure" or "angle measure", since the f ∗ (λ)-measure of an arc in S 1 is precisely its arc length (or, equivalently, the angle that it subtends at the centre of the circle.) The previous example extends nicely to give a natural "Lebesgue measure" on the n-dimensional torus T n.
This example not only generalizes the previous example, but includes many measures on non-locally compact spaces, such as Wiener measure on the space of real-valued continuous functions on the interval [0, 1]. A measure on ℝ is a Radon measure if and only if it is a locally finite Borel measure. [5] The following are not examples of Radon ...