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In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable.
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 σ {\displaystyle \sigma } -algebra) [ 3 ]
The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem. Function f is almost surely separably valued (or essentially separably valued ) if there exists a subset N ⊆ X with μ ( N ) = 0 such that f ( X \ N ) ⊆ B is separable.
Countable additivity of a measure : The measure of a countable disjoint union is the same as the sum of all measures of each subset.. Let be a set and a σ-algebra over . A set function from to the extended real number line is called a measure if the following conditions hold:
For a function f with values in a Banach space (or Fréchet space), strong measurability usually means Bochner measurability.. However, if the values of f lie in the space (,) of continuous linear operators from X to Y, then often strong measurability means that the operator f(x) is Bochner measurable for each fixed x in the domain of f, whereas the Bochner measurability of f is called uniform ...
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space.A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space.
If (,) is a measurable space and is a Banach space over a field (which is the real numbers or complex numbers), then : is said to be weakly measurable if, for every continuous linear functional:, the function : (()) is a measurable function with respect to and the usual Borel -algebra on .
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.