Complex finite-simple and smooth compact-support density for finite p

By F1, $u,v\in L^p(\mu ;\mathbb{R})$. F2 supplies real simple $a,b$ with $\mathrm{\parallel }u-a{\mathrm{\parallel }}_p<\eta \mathrm{/}2$ and $\mathrm{\parallel }v-b{\mathrm{\parallel }}_p<\eta \mathrm{/}2$, each with finite-measure nonzero set. The finite intersections of their fibers form a finite measurable partition on which $s=a+ib$ is constant, and $\{s\mathrm{\ne }0\}\subseteq \{a\mathrm{\ne }0\}\cup \{b\mathrm{\ne }0\}$ has finite measure. F1 gives $\mathrm{\parallel }f-s{\mathrm{\parallel }}_p<\eta$. This uses only two approximation choices for the specified tolerance, not a simultaneous choice function.

In Euclidean Lebesgue space, under the countable-choice hypothesis F5, apply F3 to $u,v$ with errors $\eta \mathrm{/}2$ to obtain $a,b\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{R})$. Their sum $a+ib$ is smooth componentwise and is supported in the union of the two compact supports, hence is complex $C_c^{\mathrm{\infty }}$. F1 again bounds its error by $\eta$. The inclusion $C_c^{\mathrm{\infty }}\subseteq C_c$ proves continuous compact-support density as well.

If a complex L-infinity class $f$ is in the closure of complex $C_c$, approximating $f$ to any positive tolerance and projecting its approximants gives, by F1, real $C_c$ approximations to both component classes. F4 therefore supplies $U,V\in C_0({\mathbb{R}}^n;\mathbb{R})$ representing them. The complex function $U+iV$ represents $f$ and vanishes at infinity: outside the union of two compact sets where the separate component errors are below $\eta \mathrm{/}2$, its modulus is below $\eta$.

Conversely, if $f$ has a $C_0$ representative $U+iV$, F4 supplies real $C_c$ approximants to $U,V$ with essential-norm errors below $\eta \mathrm{/}2$. Their complex sum lies in $C_c$ and has error below $\eta$ by F1. Thus precisely the stated classes form the closure. The constant-one class is excluded: any continuous representative equal to one a.e. must equal one everywhere, since a nonzero continuous discrepancy persists on an open ball of positive Lebesgue measure; that constant does not vanish at infinity. The finite-p assertions therefore have no such infinity extension.