Bochner dominated convergence theorem

Select simultaneous strong-measurability witnesses. [given, A1, L2, choose] Use [A1] exactly once to choose, for every $n$, a simple approximation sequence witnessing the strong measurability in [L2]. Unite their exceptional null sets with those from convergence and domination; countable additivity makes the union null. Modify all functions and approximants to be zero there. We now have pointwise convergence everywhere and a doubly indexed family of simple approximants.

Obtain a common countable range and measurable distances. [L3, step 1.1] Let $D$ be $\{0\}$ together with all values of all selected simple approximants. It is countable. For each $n$, the range of $f_n$ lies in $\bar{D}$, hence the pointwise limit $f$ also takes values in $\bar{D}$. For $y\in D$, the functions $\mathrm{\parallel }{f}_n-y\mathrm{\parallel }$ are measurable by applying [L3] to the simple approximants, and $\mathrm{\parallel }f-y\mathrm{\parallel }$ is measurable by applying [L3] once more to $f_n\to f$.

Build finite-valued approximants to the limit. [L2, step 2.1, construct] Enumerate $D$ with repetitions as $(y_j)$. For each $m$, assign $s_m(\omega )$ to be the least-indexed nearest point to $f(\omega )$ among $y_1,\dots ,y_m$. The finitely many tie-broken Voronoi cells are measurable by step 2.1, so $s_m$ is simple; density gives $s_m(\omega )\to f(\omega )$. Thus $f$ is strongly measurable.

Apply the scalar dominated-convergence theorem. [L1, L4, step 3.1] Passing to the pointwise limit in $\mathrm{\parallel }{f}_n\mathrm{\parallel }\le g$ gives $\mathrm{\parallel }f\mathrm{\parallel }\le g$. By [L1], $f$ and every $f_n$ are Bochner integrable. Moreover $\mathrm{\parallel }{f}_n-f\mathrm{\parallel }\le 2g$, so [L4] gives $\int \mathrm{\parallel }{f}_n-f\mathrm{\parallel }\to 0$.

Pass from $L^1$ convergence to integral convergence. [L1, L5, L6, step 4.1] Combining simple approximations to $f_n$ and $f$, [L6] and approximation independence from [L1] show that $\int (f_n-f)=\int f_n-\int f$. Apply [L5]: $\mathrm{\parallel }\int f_n-\int f\mathrm{\parallel }\le \int \mathrm{\parallel }{f}_n-f\mathrm{\parallel }\to 0$ by step 4.1. If the measure space is empty or $g=0$ a.e., every integral is zero; no separate endpoint convention is needed.