Bochner Integral: Axioms
Solution 1:
One can choose an appropriate sequence: $$\|F-S_n\|\downarrow0$$ This way one obtains a dominant: $$\|F-S_n\|\leq\|F-S_0\|:\quad\int\|F-S_0\|\mathrm{d}\mu\leq\int\|F\|\mathrm{d}\mu+\int\|S_0\|\mathrm{d}\mu<\infty+\infty$$ Thus by dominated convergence and linearity: $$\int F\mathrm{d}\mu-\int S_n\mathrm{d}\mu=\int(F-S_n)\mathrm{d}\mu\to0$$ (For more details see: Bochner Integral: Integrability)