Bochner integrability criterion

Prove necessity of scalar norm integrability. [given, L1, L2] Suppose first that $f$ is Bochner integrable and choose $(s_n)$ as in [L1]. For some $n$, $\int \mathrm{\parallel }f-s_n\mathrm{\parallel }<\mathrm{\infty }$, while integrability of the simple function gives $\int \mathrm{\parallel }{s}_n\mathrm{\parallel }<\mathrm{\infty }$. Since $\mathrm{\parallel }f\mathrm{\parallel }\le \mathrm{\parallel }f-s_n\mathrm{\parallel }+\mathrm{\parallel }{s}_n\mathrm{\parallel }$, [L2] gives $\int \mathrm{\parallel }f\mathrm{\parallel }<\mathrm{\infty }$.

Prove approximation independence. [given, L1] If $(s_n)$ and $(t_n)$ are any two defining approximations, the simple norm inequality gives $\mathrm{\parallel }\int s_n-\int t_n\mathrm{\parallel }\le \int \mathrm{\parallel }{s}_n-t_n\mathrm{\parallel }$, which is at most $\int \mathrm{\parallel }{s}_n-f\mathrm{\parallel }+\int \mathrm{\parallel }f-t_n\mathrm{\parallel }$. Both terms tend to zero, so the two norm limits coincide.

Construct dominated simple approximants for sufficiency. [given, L3, construct] Conversely assume $\int \mathrm{\parallel }f\mathrm{\parallel }<\mathrm{\infty }$. On the exceptional measurable null set of a strong approximation, replace both $f$ and every approximant by zero (and call the representative again $f$). Thus measurable simple functions $u_n$ converge pointwise to $f$; [L3] makes $\mathrm{\parallel }f\mathrm{\parallel }$ measurable. Define $s_n=u_n{\mathbf{1}}_{\{\mathrm{\parallel }{u}_n\mathrm{\parallel }\le 2\mathrm{\parallel }f\mathrm{\parallel }\}}$. Then $s_n$ is simple and measurable, $\mathrm{\parallel }{s}_n\mathrm{\parallel }\le 2\mathrm{\parallel }f\mathrm{\parallel }$, and $s_n\to f$ pointwise: when $f(\omega )\mathrm{\ne }0$, the inequality defining the retained part holds eventually, while at a zero of $f$ either retained values tend to zero or the replacement is zero.

Verify that the constructed simple functions are integrable. [L2, step 1.3] Each $s_n$ is integrable. Indeed, if a nonzero value $x$ occurs, its level set is contained in $\{2\mathrm{\parallel }f\mathrm{\parallel }\ge \mathrm{\parallel }x\mathrm{\parallel }\}$, whose measure is at most $2\int \mathrm{\parallel }f\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }x\mathrm{\parallel }<\mathrm{\infty }$ by [L2].

Obtain convergence in $L^1$. [L1, L4, step 1.3, step 2.1] The pointwise convergence in step 1.3 and $\mathrm{\parallel }f-s_n\mathrm{\parallel }\le 3\mathrm{\parallel }f\mathrm{\parallel }$ allow [L4] to be applied. Hence $\int \mathrm{\parallel }f-s_n\mathrm{\parallel }\to 0$. Together with step 2.1 this is exactly the approximation required in [L1].

Step 1.1 proves necessity, step 3.1 proves sufficiency, and step 1.2 proves that the resulting integral is approximation-independent. The zero function, the empty measure space, and a single simple function are included by taking the constant zero or constant simple approximation.