Every L^1 function admits dominated complex simple approximations

Write $f=u+iv$ with $u=\mathrm{Re}f$ and $v=\mathrm{Im}f$. [L1, L2, L3] By [L1], the real functions $u$ and $v$ are measurable and satisfy $\mathrm{\mid }u\mathrm{\mid }\le \mathrm{\mid }f\mathrm{\mid }$ and $\mathrm{\mid }v\mathrm{\mid }\le \mathrm{\mid }f\mathrm{\mid }$, hence are integrable. By [L2] and [L3], the four functions $u^{\pm }$ and $v^{\pm }$ are nonnegative measurable.

Apply [L4] to choose increasing nonnegative simple functions [L2, L3, L4, step 1.1] $u_n^{\pm }\uparrow u^{\pm }$ and $v_n^{\pm }\uparrow v^{\pm }$. Put $u_n:=u_n^{+}-u_n^{-},v_n:=v_n^{+}-v_n^{-},s_n:=u_n+i{v}_n\mathrm{.}$ Then each $s_n$ is a complex simple function, and $\mathrm{\mid }{s}_n\mathrm{\mid }\le u_n^{+}+u_n^{-}+v_n^{+}+v_n^{-}\le \mathrm{\mid }u\mathrm{\mid }+\mathrm{\mid }v\mathrm{\mid }\le 2\mathrm{\mid }f\mathrm{\mid }\mathrm{.}$

By [L5], the four increasing simple approximations in step 2.1 satisfy [L2, L5, step 2.1] $\int u_n^{\pm }d\mu \to \int u^{\pm }d\mu$ and $\int v_n^{\pm }d\mu \to \int v^{\pm }d\mu$. Therefore $\int \mathrm{\mid }u-u_n\mathrm{\mid }d\mu =\int (u^{+}-u_n^{+})d\mu +\int (u^{-}-u_n^{-})d\mu \to 0,$ and similarly $\int \mathrm{\mid }v-v_n\mathrm{\mid }d\mu \to 0$. Hence $\int \mathrm{\mid }f-s_n\mathrm{\mid }d\mu \le \int \mathrm{\mid }u-u_n\mathrm{\mid }d\mu +\int \mathrm{\mid }v-v_n\mathrm{\mid }d\mu \to 0.$

Steps 2.1 and 3.1 give the required dominated complex simple [step 2.1, step 3.1] ∎ approximations.