Bochner integral

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.

An ordinary time integral of random variables taken in the L2 space of the underlying probability space. Under strong measurability and ${\int }_a^b(\mathbb{E}|X_t{|}^2{)}^{1/2}dt<\infty$, it exists as a Bochner integral; for a mean-square continuous process it is also the limit of Riemann sums in that space. If the process is centered and the covariance is integrable, Fubini's theorem givesFor a Gaussian process, the integral is a Gaussian random variable, by approximating it in mean square by finite linear combinations of jointly Gaussian random variables.

For a probability measure whose identity map is strongly measurable with integrable norm, its barycenter is the Bochner integral of that map. It is characterized by $\phi (b(\mu ))=\int \phi (x)d\mu (x)$ for every continuous linear functional. For a measure on a weakly compact set in a separable Banach space, its barycenter belongs to the norm-closed convex hull of that set.

A function into a Banach space is strongly measurable if, outside a null set, it is a pointwise norm limit of measurable simple functions. A measurable function with values in a separable Banach space is strongly measurable: approximate values by a countable collection of balls of shrinking radii and then truncate the resulting countably valued approximants to simple functions. A strongly measurable function is Bochner integral integrable exactly when its norm is integrable. Finite second moments under a probability measure therefore imply existence of its Bochner integral expected value by the Cauchy-Schwarz inequality.