Definition:Measurable Function/Banach Space Valued Function
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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 $t \in I$.
Sources
- 2011: Wolfgang Arendt, Charles J.K. Batty, Matthias Hieber and Frank Neubrander: Vector-valued Laplace Transforms and Cauchy Problems (2nd ed.) ... (previous) ... (next): $1.1$: The Bochner Integral