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Sullivan.t.j (talk | contribs) →Definition: Added the corresponding definition (special case) of a weak random vector. |
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is a measurable function with respect to Σ and the usual [[Borel sigma algebra|Borel ''σ''-algebra]] on '''K'''.
A measurable function on a [[probability space]] is usually referred to as a [[random variable]] (or [[random vector]] if it takes values in a vector space such as the Banach space ''B'').
Thus, as a special case of the above definition, if (Ω, Σ, '''P''') is a probability space, then a function ''Z'': : Ω → ''B'' is called a (''B''-valued) '''weak random variable''' (or '''weak random vector''') if, for every continuous linear functional ''g'' : ''B'' → '''K''', the function
:<math>g \circ Z \colon \Omega \to \mathbf{K} \colon \omega \mapsto g(Z(\omega))</math>
is a '''K'''-valued random variable (i.e. measurable function) in the usual sense, with respect to Σ and the usual Borel ''σ''-algebra on '''K'''.
==Properties==
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