Random Element

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



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

      Sometimes random elements with values in [math]\displaystyle{ E }[/math] are called [math]\displaystyle{ E }[/math] -valued random variables.

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

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