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For example, for ,, or other topological spaces, the Borel algebra (generated by all the open sets) is a common choice. Some authors define measurable functions as exclusively real-valued ones with respect to the Borel algebra. [1]
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.
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 ...
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 ...
The common example for a non-measurable set which is a projection of a measurable set, is in Lebesgue 𝜎-algebra. Let be Lebesgue 𝜎-algebra of and ...
Vitali sets are examples of sets that are not measurable with respect to the Lebesgue measure. Their existence relies on the axiom of choice. Osgood curves are simple plane curves with positive Lebesgue measure [7] (it can be obtained by small variation of the Peano curve construction). The dragon curve is another unusual example.
For example, Lebesgue measure on the reals is not a probability measure, yet every universally measurable set is Lebesgue measurable. To see this, divide the real line into countably many intervals of length 1; say, N 0 =[0,1), N 1 =[1,2), N 2 =[-1,0), N 3 =[2,3), N 4 =[-2,-1), and so on. Now letting μ be Lebesgue measure, define a new measure ...