Definition 2.1.1 (Bochner Integral).label Let $(X,\cali{M},\mu)$ be a complete measure space and $E$ a normed space over $F\in\curl{\R,\com}$. For any $(\cali{M},\cali{B}(E))$-measurable functions $f,g:X\to E$ we denote $f\sim g$ if $f=g$ a.e. and define

then $\norm{\cd}{L^1(X;E)}$ is a well-defined norm on the equivalence classes of $\sim$. If $E$ is a Banach space, then

is a Banach space over $F$, called the space of Bochner integrable functions.

For each $f\in L^{1}(X;E)$, there exists $\curl{f_n}^{\infty}_{n=1}\suf L^{1}(X;E)$ taking finitely many values (or finite-valued) with $\norm{f_n(x)}{E}\leq \norm{f(x)}{E}$ for each $n\in\N$ and a.e. $x\in X$, such that $f_{n}\to f$ a.e. and in $L^{1}(X;E)$. Moreover, there exists a unique bounded linear map $I:L^{1}(X;E)\to E$ such that

1. (1)

   $I(x\cd 1_{A})=x\cd \mu(A)$ for all $x\in E$ and $A\in\cali{M}$ with $\mu(A)<\infty$

2. (2)

   $\norm{I(f)}{E}\leq \integral{X}{}\norm{f}{E}d\mu$ for all $f\in L^{1}(X;E)$