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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 ...
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean n-spaces. For lower dimensions n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume.
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.
Compactness measure is a numerical quantity representing the degree to which a shape is compact. The circle and the sphere are the most compact planar and solid shapes, respectively. The circle and the sphere are the most compact planar and solid shapes, respectively.
Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., () = for every Borel measurable set, where is the Borel measure described above). This idea extends to finite-dimensional spaces R n {\displaystyle \mathbb {R} ^{n}} (the Cramér–Wold theorem , below) but does not hold, in general, for infinite-dimensional spaces.
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.
A probability measure mapping the σ-algebra for events to the unit interval.. The requirements for a set function to be a probability measure on a σ-algebra are that: . must return results in the unit interval [,], returning for the empty set and for the entire space.