Definition:Measurable Function/Banach Space Valued Function

Definition
Let $\GF \in \set {\R, \C}$.

Let $I$ be a real interval.

Let $X$ be a Banach space over $\GF$.

Let $f : I \to X$ be a function.

We say that $f$ is measurable if there exists a sequence of simple functions $\sequence {f_n}_{n \mathop \in \N}$ such that:


 * $\ds \map f t = \lim_{n \mathop \to \infty} \map {f_n} t$

for Lebesgue almost all $t \in I$.