Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion of integral. But Bochner and Lebesgue agree on complex measurable functions. So the question arises:
Is Lebesgue obsolete? Or are there some important aspects one will miss not introducing it?
As an example, suppose that $F : [0,1]\rightarrow X$, where $X$ is a Hilbert space. Suppose that $\|F(t)\|$ is a Lebesgue integrable function. If $(F(t),x)$ is measurable for all $x$, then there is a unique vector--say $\int_{a}^{b}F\,dt$--such that $$ \int_{0}^{1}(F(t),x)\,dt = \left(\int_{0}^{1}F\,dt,x\right),\;\;\; x \in X. $$ This is a very simple integral to define, quite intuitive, powerful, does not require separability, and reduces to the scalar case.
If $X$ is not separable, then I seem to recall that the Bochner integral won't allow you to integrate general such things because $F$ may not be Bochner measurable. But you can definitely see how this might be useful, especially knowing how the integral reduces to scalar cases.