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.
Read more about this topic: Random Element
Famous quotes containing the word definition:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)