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A Hilbert-space-valued random variable is a measurable map into a <Hilbert space>. It is square-integrable when $\mathbb E\lVert X\rVert^2<\infty$, in which case its mean is defined by the <Bochner integral>.

= Random element of a Hilbert space
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Hilbert-space-valued random variable

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The Bochner integral integrates a strongly measurable function taking values in a <Banach space>. It is obtained as a norm limit of integrals of simple vector-valued functions and satisfies the usual linearity and norm bound.


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 Incoming links: Bochner integral