Bochner integral

Let $X$ be a Banach space. For an integrable simple function $s=\sum_jx_j\mathbf1_{E_j}$, set $\int s\,d\mu=\sum_jx_j\mu(E_j)$. A strongly measurable function $f$ is Bochner integrable if it admits such simple functions $s_n$ with

Its integral is $\lim_n\int s_n\,d\mu$ in $X$. The estimate $\|\int s\|\le\int\|s\|$ shows that this limit exists and is independent of the approximations.