Strongly measurable Banach-valued function

On a measure space $X$ , a function $f:X\to B$ , with $B$ a Banach space, is strongly measurable if there are measurable simple functions $s_n:X\to B$ such that $\|s_n(x)-f(x)\|_B\to0$ outside a measurable null set. A Banach-valued simple function has finite range and is written $\sum_jb_j\mathbf1_{E_j}$ with measurable $E_j$ .

Integral norms

Strong measurability makes $x\mapsto\|f(x)\|_B$ measurable up to null-set modification, as a pointwise limit of the measurable scalar norms of the approximations. If the measure is not complete, work with a measurable representative or its completion. Together with integrability of this norm, it characterizes Bochner integrability.