Completeness of the complex Haar L1 and L2 spaces and density of Cc

Discharge the Dependent Choice hypothesis of [F3] from AC. Let $R$ be entire on a nonempty set $X$ and let $a\in X$. The family $\{R[x]:x\in X\}$ of nonempty sets has, by [A1], a choice function $c$ with $c(R[x])\in R[x]$ for every $x\in X$; put $s(x):=c(R[x])$, so that $xRs(x)$ for every $x\in X$. Applying [F5] to the set $X$, the point $a$ and the function $s$ gives $g:\mathbb{N}\to X$ with $g(0)=a$ and $g(n+1)=s(g(n))$; then $g(n)Rg(n+1)$ for every $n$. This is exactly the conclusion required by [F6], so DC holds.

Discharge the countable choice hypothesis of [F2] from AC. Let $(X_n{)}_{n\in \mathbb{N}}$ be a family of nonempty sets. Its range $\{X_n:n\in \mathbb{N}\}$ is a family of nonempty sets, so by [A1] there is a choice function $c$ on that family with $c(X_n)\in X_n$ for every $n$; the function $f:\mathbb{N}\to {\bigcup }_n{X}_n$, $f(n):=c(X_n)$, satisfies $f(n)\in X_n$ for every $n$, which is the conclusion required by [F7]. So ${\mathrm{A}\mathrm{C}}_{\omega }$ holds.

For every $z\in \mathbb{C}$ one has $\mathrm{\mid }\mathrm{Re}z\mathrm{\mid }\le \mathrm{\mid }z\mathrm{\mid }$, $\mathrm{\mid }\mathrm{Im}z\mathrm{\mid }\le \mathrm{\mid }z\mathrm{\mid }$ and $\mathrm{\mid }z\mathrm{\mid }\le \mathrm{\mid }\mathrm{Re}z\mathrm{\mid }+\mathrm{\mid }\mathrm{Im}z\mathrm{\mid }$; consequently for all $z,w\in \mathbb{C}$ also $\mathrm{\mid }\mathrm{Re}z-\mathrm{Re}w\mathrm{\mid }=\mathrm{\mid }\mathrm{Re}(z-w)\mathrm{\mid }\le \mathrm{\mid }z-w\mathrm{\mid }$ and $\mathrm{\mid }\mathrm{Im}z-\mathrm{Im}w\mathrm{\mid }\le \mathrm{\mid }z-w\mathrm{\mid }$, and for real $a,b$ one has $(\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }{)}^2\le 2\mathrm{\mid }a{\mathrm{\mid }}^2+2\mathrm{\mid }b{\mathrm{\mid }}^2$.

Let $(f_n)$ be a Cauchy sequence in $L^1(X,\mu ;\mathbb{C})$. Its real and imaginary parts are Cauchy in the real space $L^1(\mu )$: by the first two inequalities of step 1.3 and monotonicity [F4], $\int \mathrm{\mid }\mathrm{Re}{f}_n-\mathrm{Re}{f}_m\mathrm{\mid }d\mu \le \int \mathrm{\mid }{f}_n-f_m\mathrm{\mid }d\mu$ and likewise for the imaginary parts, the two left-hand sides being the norms of the real classes $\mathrm{Re}{f}_n-\mathrm{Re}{f}_m$ and $\mathrm{Im}{f}_n-\mathrm{Im}{f}_m$. The same argument with $\mathrm{\mid }\cdot {\mathrm{\mid }}^2$ in place of $\mathrm{\mid }\cdot \mathrm{\mid }$ and the last inequality of step 1.3 shows that a Cauchy sequence in $L^2(X,\mu ;\mathbb{C})$ has real and imaginary parts that are Cauchy in the real space $L^2(\mu )$.

Complex density. Let $f\in L^p(X,\mu ;\mathbb{C})$ with $p\in \{1,2\}$ and let $ϵ>0$. The real and imaginary parts of $f$ are real classes in $L^p(\mu )$ by [F1], and $p$ is finite, so by [F3] under the Dependent Choice of step 1.1 there are real $u,v\in C_c(X)$ with $\mathrm{\parallel }\mathrm{Re}f-u{\mathrm{\parallel }}_p<ϵ\mathrm{/}2$ and $\mathrm{\parallel }\mathrm{Im}f-v{\mathrm{\parallel }}_p<ϵ\mathrm{/}2$. Then $u+iv\in C_c(X;\mathbb{C})$ by [F8], and for $p=1$ steps 1.3 and [F4] give $\mathrm{\parallel }f-(u+iv){\mathrm{\parallel }}_1\le \mathrm{\parallel }\mathrm{Re}f-u{\mathrm{\parallel }}_1+\mathrm{\parallel }\mathrm{Im}f-v{\mathrm{\parallel }}_1<ϵ$, while for $p=2$ the last inequality of step 1.3 and [F4] give $\mathrm{\parallel }f-(u+iv){\mathrm{\parallel }}_2^2=\int \mathrm{\mid }(\mathrm{Re}f-u)+i(\mathrm{Im}f-v){\mathrm{\mid }}^{2}d\mu \le 2\mathrm{\parallel }\mathrm{Re}f-u{\mathrm{\parallel }}_2^2+2\mathrm{\parallel }\mathrm{Im}f-v{\mathrm{\parallel }}_2^2<{ϵ}^2$. Thus $C_c(X;\mathbb{C})$ is dense in $L^p(X,\mu ;\mathbb{C})$ for both $p$.

Let $(f_n)$ be a Cauchy sequence in $L^1(X,\mu ;\mathbb{C})$, or in $L^2(X,\mu ;\mathbb{C})$; fix $p\in \{1,2\}$ accordingly. By step 2.1 the real sequences $(\mathrm{Re}{f}_n)$ and $(\mathrm{Im}{f}_n)$ are Cauchy in the real space $L^p(\mu )$, so by [F2] under the countable choice of step 1.2 there are $g,h\in L^p(\mu )$ with $\mathrm{\parallel }\mathrm{Re}{f}_n-g{\mathrm{\parallel }}_p\to 0$ and $\mathrm{\parallel }\mathrm{Im}{f}_n-h{\mathrm{\parallel }}_p\to 0$.

Recombination, for $p\in \{1,2\}$ as in step 3.1. Define $f:=g+ih$, the class of the measurable complex function $g+ih$, which lies in $L^p(X,\mu ;\mathbb{C})$ because $\mathrm{\mid }f\mathrm{\mid }\le \mathrm{\mid }g\mathrm{\mid }+\mathrm{\mid }h\mathrm{\mid }$ and $\mathrm{\mid }g\mathrm{\mid }+\mathrm{\mid }h\mathrm{\mid }\in L^p(\mu )$ for the finite $p$ by steps 1.3 and [F4] in the case $p=1$ and by the last inequality of step 1.3 and [F4] in the case $p=2$. Then $\mathrm{\parallel }{f}_n-f{\mathrm{\parallel }}_p\to 0$ by the same two computations applied to $f_n-f=(\mathrm{Re}{f}_n-g)+i(\mathrm{Im}{f}_n-h)$ together with step 3.1, so $(f_n)$ converges in $L^p(X,\mu ;\mathbb{C})$. Hence both complex spaces are complete.

Combining the density of step 2.2 with the completeness of step 4.1, the complex spaces $L^1(X,\mu ;\mathbb{C})$ and $L^2(X,\mu ;\mathbb{C})$ are complete and $C_c(X;\mathbb{C})$ is dense in both. ∎