 Hilbert-space-valued random variable

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 $\mathbb{E}\Vert X\Vert$ defines its mean by the Bochner integral; finite $\mathbb{E}\Vert X{\Vert }^2$ 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.