Random element

In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet (1948) who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.”

The modern day usage of “random element” frequently assumes the space of values is a topological vector space, often a Banach or Hilbert space with a specified natural sigma algebra of subsets.

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

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