Random Element - Definition

Definition

Let (Ω, ℱ, P) be a probability space, and (E, ℰ) a measurable space. A random element with values in E is a function X: Ω→E which is (ℱ, ℰ)-measurable. That is, a function X such that for any B ∈ ℰ the preimage of B lies in ℱ: {ω: X(ω) ∈ B} ∈ ℱ.

Sometimes random elements with values in are called -valued random variables.

Note if, where are the real numbers, and is its Borel σ-algebra, then the definition of random element is the classical definition of random variable.

The definition of a random element with values in a Banach space is typically understood to utilize the smallest -algebra on B for which every bounded linear functional is measurable. An equivalent definition, in this case, to the above, is that a map, from a probability space, is a random element if is a random variable for every bounded linear functional f, or, equivalently, that is weakly measurable.