Definition:Simple Function/Banach Space

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 simple there exists:


 * Lebesgue measurable subsets $\Omega_1, \ldots, \Omega_r$ of $I$ with finite Lebesgue measure
 * $x_1, \ldots, x_r \in X$

such that:


 * $\ds \map f t = \sum_{r \mathop = 1}^n x_r \map {\chi_{\Omega_r} } t$

for each $t \in I$.