Let
- $(\Omega,\mathcal A)$ be a measurable space
- $E$ be a $\mathbb R$-Banach space
- $\mu:\mathcal A\to E$ be $\sigma$-additive and of bounded variation
- $F,G$ be $\mathbb R$-Banach spaces
- $\mathfrak b:E\times F\to G$ be a bounded bilinear operator
If $f=\sum_{i=1}^k1_{A_i}x_i$ is a $\mathcal A$-elementary function $\Omega\to E$, let $$\int f\:{\rm d}\mu:=\sum_{i=1}^k\mathfrak b(\mu(A_i),f_i).$$
Now, let $|\mu|:\mathcal A\to[0,\infty)$ denote the variation of $\mu$ and $$\mathcal L^1(\mu;F):=\mathcal L^1(|\mu|;F).$$ It's easy to see that $$\left\|\int f\:{\rm d}\mu\right\|_G\le\left\|\mathfrak b\right\|\left\|f\right\|_{\mathcal L^1(\mu;\:F)}.\tag1$$ Thus, $$f\mapsto\int f\:{\rm d}\mu\tag2$$ is a bounded linear operator from the space of $\mathcal A$-elementary functions equipped with $\left\|\;\cdot\;\right\|_{\mathcal L^1(\mu;\:F)}$ to $G$ and hence has a unique extension to a bounded linear operator from $\mathcal L^1(\mu;F)$ to $G$ with $(1)$.
In that way, we've defined an integral $$\int f\:{\rm d}\mu\tag3$$ for all $f\in\mathcal L^1(\mu;F)$.
How is this integral called? And should we use the notation $(3)$ for it?
My problem with $(3)$ is that it does not stress the fact that the integral has to be understood with respect to $\mathfrak b$.
It has to be understood with respect to $\mathfrak{b}$: Given two distinct $\mathfrak{b}$ and $\mathfrak{b'}$, then the approximate sequences to $(\mathfrak{b})\displaystyle\int fd\mu$ and $(\mathfrak{b'})\displaystyle\int fd\mu$ are different that $\displaystyle\sum_{i=1}^{k}\mathfrak{b}(\mu(A_{i}),f_{i})\ne\sum_{i=1}^{k}\mathfrak{b'}(\mu(A_{i}),f_{i})$ in general.
I suspect it is also called as Bochner integrals. For a good reference to all these issues, I will recommend Vector Measures, N. Dinculeanu.