Complex Lp completeness and almost-everywhere subsequences

For $u_n=\mathrm{Re}[f_n]$ and $v_n=\mathrm{Im}[f_n]$, F1 gives $\mathrm{\parallel }{u}_n-u_m{\mathrm{\parallel }}_p,\mathrm{\parallel }{v}_n-v_m{\mathrm{\parallel }}_p\le \mathrm{\parallel }[f_n]-[f_m]{\mathrm{\parallel }}_p$. Both real sequences are therefore Cauchy. F2 supplies real classes $u,v$ with $u_n\to u$, $v_n\to v$.

For a sequence already converging to $[f]$, its real components converge to $\mathrm{Re}[f]$ by F1. Apply F3 to obtain indices $n_k$ and real representatives $a_k\to U$ off a measurable null set. Its imaginary components still converge in norm; apply F3 to that subsequence to obtain further indices $k_j$ and imaginary representatives $b_j\to V$ off a second measurable null set. Then $a_{k_j}+i{b}_j$ represents $[f_{n_{k_j}}]$ and converges to $U+iV$ outside the union of those two null sets, which is null by F5.

Choose measurable representatives $U,V$ of these two classes and put $f=U+iV$. F1 shows $f\in {\mathcal{L}}^p(\mu ;\mathbb{C})$ and $\mathrm{\parallel }[f_n]-[f]{\mathrm{\parallel }}_p\le \mathrm{\parallel }{u}_n-u{\mathrm{\parallel }}_p+\mathrm{\parallel }{v}_n-v{\mathrm{\parallel }}_p\to 0$. Thus every Cauchy sequence converges, at infinity as well as at finite $p$.

The simultaneous representative selections used by the real results are permitted by F4; selection of the two limit representatives requires only two choices. All functions can be assigned zero on the measurable exceptional sets: a function pieced from a measurable function on a measurable set and zero on its complement is measurable. If predetermined measurable representatives are desired, their disagreement sets with the selected representatives are themselves measurable and null; F5 applied to their countable union preserves the a.e. convergence. Hence the assertions hold on incomplete measures without prescribing arbitrary, possibly nonmeasurable, values on null sets.