# 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 $\displaystyle{ (\Omega, \mathcal{F}, P) }$ be a probability space, and $\displaystyle{ (E, \mathcal{E}) }$ a measurable space. A 'random element with values in E is a function 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 $\displaystyle{ \mathcal{F} }$ .

Sometimes random elements with values in $\displaystyle{ E }$ are called $\displaystyle{ E }$ -valued random variables.

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

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