= Strongly measurable function
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>.
Back to article page
