Existence of an injective measurable function from \ textstyle ( \ Omega, \ mathcal { F }, P ) to a standard measurable space \ textstyle ( X, \ Sigma ) does not depend on the choice of \ textstyle ( X, \ Sigma ).
32.
The result is important to classical Banach space theory, in that, when considering the Banach space given as an " L " " p " space of measurable functions over a general measurable space, it is sufficient to understand it in terms of its decomposition into non-atomic and atomic parts.
33.
The measurable space ( \ Omega, \ mathcal F ) is said to have the "'regular conditional probability property "'if for all probability measures P on ( \ Omega, \ mathcal F ), all random variables on ( \ Omega, \ mathcal F, P ) admit a regular conditional probability.
34.
In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function " X " from a probability space \ scriptstyle ( \ Omega, \ mathcal { F }, \ operatorname { P } ) to measurable space \ scriptstyle ( \ mathcal { X }, \ mathcal { A } ).
35.
In quantum mechanics, the unit sphere of the Hilbert space " H " is interpreted as the set of possible states ? of a quantum system, the measurable space " X " is the value space for some quantum property of the system ( an " observable " ), and the projection-valued measure ? expresses the probability that the observable takes on various values.
36.
"' Terminological note "': The terminology adopted by the literature on the subject is followed here, according to which a measurable space " X " is referred to as a " Borel space " and the elements of the distinguished ?-algebra of " X " as Borel sets, regardless of whether or not the underlying ?-algebra comes from a topological space ( in most examples it does ).
