# Random Element

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A Random Element is a measurable real-valued function that ...

## References

### 2015

1. V.V. Buldygin, A.B. Kharazishvili. Geometric Aspects of Probability Theory and Mathematical Statistics. – Kluwer Academic Publishers, Dordrecht. – 2000

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/random_element#Definition Retrieved:2015-5-16.
• Let $(\Omega, \mathcal{F}, P)$ be a probability space, and $(E, \mathcal{E})$ a measurable space. A 'random element with values in E is a function which is $(\mathcal{F}, \mathcal{E})$ -measurable. That is, a function X such that for any $B\in \mathcal{E}$, the preimage of B lies in $\mathcal{F}$ .

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

Note if $(E, \mathcal{E})=(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ , where $\mathbb{R}$ are the real numbers, and $\mathcal{B}(\mathbb{R})$ is its Borel σ-algebra, then the definition of random element is the classical definition of random variable.

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