= Hilbert-space-valued random variable
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= Random element of a Hilbert space
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{synonym}
A <Hilbert-space-valued random variable> is a measurable map into a <Hilbert space>, equipped with its <Borel sigma-algebra>. For a separable real <Hilbert space>, finite $\mathbb E\|X\|$ defines its mean by the <Bochner integral>; finite $\mathbb E\|X\|^2$ also defines a <covariance operator>. A random square-integrable <function> on $[0,1]$, viewed as an element of <L2 space>, is an example. Its values are whole equivalence classes of <functions>, so point evaluation is not automatically meaningful. In a nonseparable space, strong measurability or an appropriate separable-valued <hypothesis> must be checked before using the same <integral> construction.
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