Linearity of the Bochner integral

The functions $f,g$ are strongly measurable by [F1]. Let $u_n,v_n$ be their measurable simple approximations converging pointwise outside measurable null sets $N,N\mathrm{\prime }$; these need not be the defining $L^1$ approximations $s_n,t_n$. The simple functions $\alpha u_n+\beta v_n$ converge to $\alpha f+\beta g$ off $N\cup N\mathrm{\prime }$ by [F4], proving strong measurability.

Norm estimate: for every $n$, $\mathrm{\parallel }\alpha f+\beta g-(\alpha s_n+\beta t_n)\mathrm{\parallel }\le \mathrm{\mid }\alpha \mathrm{\mid }\mathrm{\parallel }f-s_n\mathrm{\parallel }+\mathrm{\mid }\beta \mathrm{\mid }\mathrm{\parallel }g-t_n\mathrm{\parallel }$ pointwise, hence after integration ${\int }_{\mathrm{\Omega }}\mathrm{\parallel }\alpha f+\beta g-(\alpha s_n+\beta t_n)\mathrm{\parallel }d\mu \le \mathrm{\mid }\alpha \mathrm{\mid }{\int }_{\mathrm{\Omega }}\mathrm{\parallel }f-s_n\mathrm{\parallel }d\mu +\mathrm{\mid }\beta \mathrm{\mid }{\int }_{\mathrm{\Omega }}\mathrm{\parallel }g-t_n\mathrm{\parallel }d\mu \to 0$; in particular ${\int }_{\mathrm{\Omega }}\mathrm{\parallel }\alpha f+\beta g\mathrm{\parallel }d\mu \le \mathrm{\mid }\alpha \mathrm{\mid }{\int }_{\mathrm{\Omega }}\mathrm{\parallel }f\mathrm{\parallel }d\mu +\mathrm{\mid }\beta \mathrm{\mid }{\int }_{\mathrm{\Omega }}\mathrm{\parallel }g\mathrm{\parallel }d\mu <\mathrm{\infty }$, since $\int \mathrm{\parallel }f\mathrm{\parallel },\int \mathrm{\parallel }g\mathrm{\parallel }<\mathrm{\infty }$ by [F3].

By [step 1.1], [step 1.2] and the integrability criterion [F3], the function $\alpha f+\beta g$ is Bochner integrable, and $\alpha s_n+\beta t_n$ is a defining sequence of integrable simple functions for it, so ${\int }_{\mathrm{\Omega }}(\alpha f+\beta g)d\mu ={\lim }_n{\int }_{\mathrm{\Omega }}(\alpha s_n+\beta t_n)d\mu$.

Linearity of the simple integral [F2] gives ${\int }_{\mathrm{\Omega }}(\alpha s_n+\beta t_n)d\mu =\alpha {\int }_{\mathrm{\Omega }}{s}_{n}d\mu +\beta {\int }_{\mathrm{\Omega }}{t}_{n}d\mu$ for every $n$, whose right-hand side converges to $\alpha {\int }_{\mathrm{\Omega }}fd\mu +\beta {\int }_{\mathrm{\Omega }}gd\mu$ by [F1] and continuity of the vector operations [F4]; combining with [step 2.1] yields ${\int }_{\mathrm{\Omega }}(\alpha f+\beta g)d\mu =\alpha {\int }_{\mathrm{\Omega }}fd\mu +\beta {\int }_{\mathrm{\Omega }}gd\mu$.

Restricted form: the functions $f{\mathbf{1}}_E$ and $g{\mathbf{1}}_E$ are Bochner integrable, because they are strongly measurable and dominated in norm by $\mathrm{\parallel }f\mathrm{\parallel }$ and $\mathrm{\parallel }g\mathrm{\parallel }$ respectively, and $(\alpha f+\beta g){\mathbf{1}}_E=\alpha (f{\mathbf{1}}_E)+\beta (g{\mathbf{1}}_E)$ pointwise; applying [step 3.1] to the pair $f{\mathbf{1}}_E,g{\mathbf{1}}_E$ gives ${\int }_E(\alpha f+\beta g)d\mu =\alpha {\int }_{E}fd\mu +\beta {\int }_{E}gd\mu$.

The integral is thus additive and homogeneous on the Bochner integrable functions: taking $\alpha =\beta =1$ gives additivity, $\beta =0$ with $\alpha$ arbitrary gives homogeneity, and $\beta =-1$ shows the difference $f-g$ is Bochner integrable with ${\int }_{\mathrm{\Omega }}(f-g)d\mu ={\int }_{\mathrm{\Omega }}fd\mu -{\int }_{\mathrm{\Omega }}gd\mu$, so the integral is well defined on differences.