Bounded linear maps commute with Bochner integration

Restrict a defining approximation to the measurable set. Choose integrable simple $s_n$ with $\int \mathrm{\parallel }f-s_n\mathrm{\parallel }\to 0$. Then ${\mathbf{1}}_{E}T{s}_n$ is an integrable $Y$-valued simple function: every nonzero level is a finite union of level sets of ${\mathbf{1}}_E{s}_n$.

Prove Bochner integrability after applying $T$. By [L1], $\int \mathrm{\parallel }{\mathbf{1}}_{E}Tf-{\mathbf{1}}_{E}T{s}_n\mathrm{\parallel }\le C{\int }_E\mathrm{\parallel }f-s_n\mathrm{\parallel }\to 0$. Thus [L2] makes ${\mathbf{1}}_{E}Tf$ Bochner integrable.

Commute $T$ with the defining limit. [L1, L2, L3, step 1.1, step 2.1] For each simple $s_n$, finite linearity in [L3] gives $T({\int }_E{s}_n)={\int }_{E}T{s}_n$. Boundedness makes $T$ norm-continuous, so taking limits in this equality and using [L2] proves the displayed identity. If $T=0$, $f=0$, or $E=\mathrm{\varnothing }$, both sides are explicitly zero.