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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 ...
Consider otherwise the space (, (),) where denotes the counting measure. This space is atomic, with all atoms being the singletons, yet the space is not able to be partitioned into the disjoint union of countably many disjoint atoms, = and a null set since the countable union of singletons is a countable set, and the uncountability of the real ...
A probability space is a measure space such that the measure of the whole space is equal to 1. The product of any family (finite or not) of probability spaces is a probability space. In contrast, for measure spaces in general, only the product of finitely many spaces is defined. Accordingly, there are many infinite-dimensional probability ...
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.
In mathematics, a measurable space or Borel space [1] is a basic object in measure theory. It consists of a set and a σ-algebra , which defines the subsets that will be measured.
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space ( X , Σ, μ ) is complete if and only if [ 1 ] [ 2 ]
The Hausdorff measure is a generalization of the Lebesgue measure that is useful for measuring the subsets of R n of lower dimensions than n, like submanifolds, for example, surfaces or curves in R 3 and fractal sets. The Hausdorff measure is not to be confused with the notion of Hausdorff dimension.