Weakly measurable function
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
—specifically, in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
—a weakly measurable function taking values in a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
is a function whose composition with any element of the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
is a
measurable function 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. This is in di ...
in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree.


Definition

If (X, \Sigma) is a measurable space and B is a Banach space over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
\mathbb (which is the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s \R or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s \Complex), then f : X \to B is said to be weakly measurable if, for every continuous linear functional g : B \to \mathbb, the function g \circ f \colon X \to \mathbb \quad \text \quad x \mapsto g(f(x)) is a measurable function with respect to \Sigma and the usual Borel \sigma-algebra on \mathbb. A measurable function on a probability space is usually referred to as a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
(or random vector if it takes values in a vector space such as the Banach space B). Thus, as a special case of the above definition, if (\Omega, \mathcal) is a probability space, then a function Z : \Omega \to B is called a (B-valued) weak random variable (or weak random vector) if, for every continuous linear functional g : B \to \mathbb, the function g \circ Z \colon \Omega \to \mathbb \quad \text \quad \omega \mapsto g(Z(\omega)) is a \mathbb-valued random variable (i.e. measurable function) in the usual sense, with respect to \Sigma and the usual Borel \sigma-algebra on \mathbb.


Properties

The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem. A function f is said to be
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
separably valued (or essentially separably valued) if there exists a subset N \subseteq X with \mu(N) = 0 such that f(X \setminus N) \subseteq B is separable. In the case that B is separable, since any subset of a separable Banach space is itself separable, one can take N above to be empty, and it follows that the notions of weak and strong measurability agree when B is separable.


See also

* * * * *


References

* * {{Analysis in topological vector spaces Functional analysis Measure theory Types of functions