Condition for separability of $L^2_C(Z,\nu)$ in Dixmier Von Neumann Algebras proof I was reading up on direct integrals of Hilbert spaces and saw the section on it from Dixmier's Von Neumann Algebras on Google Books. In Part II Chapter 1 Section 6 "Basic Properties of Direct Integrals", the Corollary to Proposition 6 reads "If $\nu$ is standard, $H$ is separable." The proof's first sentence reads "By the hypothesis on $\nu$, there exists a sequence $(f_1,f_2,\dots)$ of complex valued functions dense in $L^2_C(Z,\nu)$." I understand how the proof goes from there, so I am really just interested in this one sentence.
First, I am assuming that $L^2_C(Z,\nu)$ is the space of equivalence classes of square-integrable (with respect to $\nu$) complex-valued functions on $Z$, where functions equal $\nu$-almost everywhere are in the same equivalence class. In other words, I assume the subscript $C$ means values in $\mathbb{C}$. Is that right?
Second, I know the definition of $\nu$ being standard is that $Z-N$ is standard for some set $N$ contained in a measurable set of $\nu$-measure $0$. And I know that a Borel space $(Z,\mathscr{B})$ is standard if $\mathscr{B}$ is the $\sigma$-algebra of Borel sets of a Polish space. Finally, I understand that a Polish space is a topological space whose topology is second countable and compatible with a complete metric.
Next, I know that Dixmier assumes that $\nu$ is countably additive and $\sigma$-finite. As an example, he states that if "$Z$ is a locally compact space [and I think he meant to include Hausdorff in that description], countable at infinity [which I understand means the same as $\sigma$-compact], a positive (Radon) measure on $Z$, regarded as a function on the set of Borel sets of $Z$, is a positive measure in the above sense. When $Z$ is second countable, this measure is standard."
Now I've seen some different definitions of a Radon measure, but as I understand it, we are just trying to get enough regularity to be able to prove some things (like the separability referenced above).
Now, I am not trying to study von Neumann algebras, or anything where I need the precision of standard spaces and Polish spaces. Nor am I interested in Bourbaki's definition of a measure on a locally compact space as a linear functional on a certain inductive limit of function spaces on compact sets.
I am just interested right now in conditions on a Borel measure space $(Z,\mathscr{B},\nu)$ such that $L^2(\nu)$ is separable. I know a proof if $Z$ is $\mathbb{R}^n$, but it requires the use of complex valued polynomials in $n$ variables. But if $Z$ is not a subset of $\mathbb{R}^n$ or even of $\mathbb{C}^n$, then that's not going to work.
So suppose I restrict $Z$ to being a second countable locally compact Hausdorff space which is $\sigma$-compact, and that $\nu$ is a $\sigma$-finite positive Borel measure on the Borel sets of $Z$. What regularity assumptions do I need to make on $\nu$ to be able to prove that $L^2(Z,\mathscr{B},\nu)$ is separable and how does the proof go? If possible, please exhibit a proof that does not depend on "known" facts about Radon measures and Polish or standard spaces. Please feel
free to assume I know that $C_c(Z)$ is dense in $L^2(\nu)$ if $\nu$ is regular. (Actually I think that one can get by with inner and outer regularity just for sets of finite measure, and maybe also that compact sets have finite measure.) Also, anything else from Rudin Real and Complex Analysis is OK.
Thanks
 A: In order to make answering the question easier for me, I am going to change the terminology a little bit. I will prove the following:
Theorem. Let $X$ be a locally compact second countable Hausdorff space, let $\mathscr{B}$ be the Borel sets of $X$, and let $\mu$ be a positive measure on $\mathscr{B}$ which has the following regularity properties:

*

*$\mu(K)<\infty$ for every compact set $K\subseteq X$.

*if $E\in\mathscr{B}$ and $\mu(E)<\infty$ then
$$\mu(E)=\inf\,\{\mu(V)\colon E\subseteq V,\text{ $V$ open}\}.$$

*if $E\in\mathscr{B}$ and $\mu(E)<\infty$ then
$$\mu(E)=\sup\,\{\mu(K)\colon K\subseteq E,\text{ $K$ compact}\}.$$
[Property (2) would normally be called outer regularity for Borel sets of finite
measure, while (3) would be called inner regularity for Borel sets of finite
measure.]

Then $L^p(\mu)=L^p(X,\mathscr{B},\mu)$ is separable for $1\leq p<\infty$.
I will frequently cite references to the following texts:

*

*KGT: Kelley, General Topology

*LANG: Lang, Real and Functional Analysis, Third Edition

*RCA: Rudin, Real & Complex Analysis, Third Edition

*RFA: Rudin, Functional Analysis, Second Edition

The key to my proof was an idea found in LANG Chapter III Section 4 Exercise 10.
Proof: Since $X$ is second countable, let $D$ be a countable base for the topology
of $X$. Let $C=\{U\in D\colon\bar{U}\text{ is compact}\}$. Then $C$ is also a
countable base, for if $x\in V$, an open subset of $X$, then $\{x\}$ is compact,
so by RCA 2.7, there exists an open set $W$ such that $\bar{W}$ is compact and
$x\in W\subseteq\bar{W}\subseteq V$. Then, for some $U\in D$,
$x\in U\subseteq W\subseteq V$, so $\bar{U}\subseteq\bar{W}$ and hence
$\bar{U}$ is compact, and therefore $U\in C$. Write $C=\{U_1,U_2,\dots\}$.
This next part is taken from the proof of LANG Chapter IX Theorem 5.3. We will
construct, inductively, a sequence of integers $0=j_1<j_2<\cdots$ and a sequence
$K_1,K_2,\dots$ of compact sets such that for $i=1,2,\dots$,
\begin{equation*}
  K_i=\bar{U}_1\cup\dots\cup\bar{U}_{j_i+1}
  \subseteq K_{i+1}^\circ\qquad(i=1,2,\dots).
\end{equation*}
Let $K_1=\bar{U}_1$.
Suppose we have constructed $j_1,\dots,j_i$ and $K_1,\dots,K_i$. Then $K_i$ is
compact and $C$ is an open cover for $K_i$. Let $j_{i+1}$ be the smallest integer
greater than $j_i$ such that $K_i\subseteq U_1\cup\dots\cup U_{j_{i+1}}$, and let
$K_{i+1}=\bar{U}_1\cup\dots\cup\bar{U}_{j_{i+1}+1}$, which is compact. Then
\begin{equation*}
  K_i\subseteq U_1\cup\dots\cup U_{j_{i+1}}\text{ open }
  \subseteq\bar{U}_1\cup\dots\cup\bar{U}_{j_{i+1}}\cup\bar{U}_{j_{i+1}+1},
\end{equation*}
so $K_i\subseteq(\bar{U}_1\cup\dots\cup\bar{U}_{j_{i+1}+1})^\circ=K_{i+1}^\circ$.
If $x\in X$ then $x\in U_k$ for some $k$. Let $i$ be such that $j_i\geq k$. Then
$x\in U_k\subseteq\bar{U}_1\cup\dots\cup\bar{U}_{j_i}\cup\bar{U}_{j_i+1}=K_i$, so
$$X=\bigcup_{i=1}^\infty K_i;$$
that is, $X$ is $\sigma$-compact.
Fix $i$ and set $S=K_i$. $S$ is a second countable compact Hausdorff space, being a
subset of $X$, which itself is second countable Hausdorff. Therefore $S$ is a $T_1$
space and so by KGT 5.9, $S$ is normal, and hence regular, since it is $T_1$.
Therefore, by KGT 4.16 (Urysohn Metrization Theorem), $S$ is metrizable. Let $d$ be a
compatible metric, and by KGT 4.13, we may assume that $d(s,t)\leq 1$ for all
$s,t\in S$. By KGT 1.14, let $\{s_1,s_2,\dots\}$ be a countable dense set of distinct
elements of $S$. For $n=1,2,\dots$, define $g_n\colon S\to\mathbb{C}$ by
$g_n(s)=d(s,s_n)$. By KGT 4.8, $g_n\in C(S)$, and we have that $0\leq g_n(s)\leq 1$
for all $s\in S$. Let $B$ be the subalgebra of $C(S)$ consisting of all polynomials
with complex coefficients in a finite number of variables, evaluated on a finite
subset of $\{g_1,g_2,\dots\}$. That is, they are polynomials in the $k$ variables
$g_{n_1},\dots,g_{n_k}$ for all possible selections of $k$ members of
$\{g_1,g_2,\dots\}$, for $k=1,2,\dots$. $B$ is self-adjoint (see RFA 5.7(b) for
terminology) as all we have to do is to take the complex conjugate of the
coefficients since the $g_n$ are all real. If $s,t\in S$ with $s\neq t$, let
$\epsilon=d(s,t)>0$. Then for some $n$, $d(s,s_n)<\epsilon/2$. Now
$\epsilon=d(s,t)\leq d(s,s_n)+d(t,s_n)$ so
$$g_n(t)=d(t,s_n)\geq\epsilon-d(s,s_n)>\epsilon/2>d(s,s_n)=g_n(s),$$
and hence $B$ separates points on $S$. If $s\in S$, then $f(s)\neq 0$ for the
constant polynomial $f=1$ in $B$. Therefore, by RFA 5.7 (Stone-Weierstrass Theorem),
$B$ is dense in $C(S)$ in the $\sup$ norm. If we let $\check{B}$ be defined just as
$B$ was, but we
restrict the coefficients to be complex numbers whose real and imaginary parts are
rational (such a number is called rational complex), then $\check{B}$ is countable.
A polynomial in $\check{B}$ of degree $N$ in $k$ variables is of the form
$$\check{p}(g)=\sum_{\lvert\alpha\rvert\leq N}q_\alpha g^\alpha,$$
where $\alpha$ is a multi-index (see RFA 1.34 for the definition),
$g=(g_{n_1},\dots,g_{n_k})$,
and $\Re q_\alpha,\Im q_\alpha\in\mathbb{Q}$.
Let $\epsilon>0$ be given along with a polynomial $p\in B$, say
$$p(g)=\sum_{\lvert\alpha\rvert\leq N}c_\alpha g^\alpha,$$
where $c_\alpha$ is a complex number for each $\alpha$ such that
$\lvert\alpha\rvert<N$ and $g=(g_{n_1},\dots,g_{n_k})$.
Then $\lvert g_{n_j}(s)\rvert\leq 1$ for all $s\in S$ and $j=1,\dots,k$, so for
$s\in S$ and $\lvert\alpha\rvert\leq N$,
\begin{equation*}
  \lvert g^\alpha(s)\rvert
  =\lvert g_{n_1}^{\alpha_1}(s)\cdots g_{n_k}^{\alpha_k}(s)\rvert\leq 1
  \qquad(s\in S,\,\lvert\alpha\rvert\leq N).
\end{equation*}
Then, if for each $\alpha$ such that $\lvert\alpha\rvert\leq N$, a rational complex number $q_\alpha$ is chosen such that
$$\lvert c_\alpha-q_\alpha\rvert<\frac{\epsilon}{(N+1)^k},$$
then for all $s\in S$,
\begin{equation*}
  \lvert(p(g))(s)-(\check{p}(g))(s)\rvert
  \leq\sum_{\lvert\alpha\rvert\leq N}\lvert
  c_\alpha-q_\alpha\rvert\,\lvert g^\alpha(s)\rvert
  <\sum_{\lvert\alpha\rvert\leq N}\frac{\epsilon}{(N+1)^k}<\epsilon,
\end{equation*}
so $\check{B}$ is dense in $B$ and hence also in $C(S)$, so $C(S)$ is separable. Thus
$C(K_i)$ is separable for $i=1,2,\dots$, with a countable dense set $\check{B}_i$
of polynomials.
Zero extend every $\check{p}\in\check{B}_i$ to a function $p^*$ on $X$.
Let $P_i=\{p^*\colon\check{p}\in\check{B}_i\}$ and put
$$P=\bigcup_{i=1}^\infty P_i.$$
Then $P$ is countable and $P\subseteq L^p(\mu)$ since $p^*$ is bounded and
$\mu(K_i)<\infty$. Let $f\in L^p(\mu)$ and let $\epsilon>0$ be given. By RCA 3.14,
$C_c(X)$ is dense in $L^p(\mu)$ [please note that the regularity conditions
on $\mu$ required by the proof of RCA 3.14 are precisely those listed as 1-3 in the
statement of the theorem], so there exists a $g\in C_c(X)$ such that
$\lvert\lvert f-g\rvert\rvert_p<\epsilon/2$. Let $K$ be the support of $g$.
Then $K$ is compact. If $x\in K\subseteq X$, then $x\in K_i\subseteq K_{i+1}^\circ$
for some $i$. Therefore $\{K_{i+1}^\circ\colon K\cap K_{i+1}^\circ\neq\varnothing\}$
is an open cover of $K$ and so
$K\subseteq K_{i_1+1}^\circ\cup\dots\cup K_{i_j+1}^\circ\subseteq K_{i_j+1}$
for some $i_1<\dots<i_j$. Hence $g|K_{i_j+1}\in C(K_{i_j+1})$. Then there exists
$\check{p}\in\check{B}_{i_j+1}$ such that
\begin{equation*}
  \sup_{s\in K_{i_j+1}}\,\lvert(g|K_{i_j+1})(s)-\check{p}(s)\rvert
  <\frac{\epsilon}{2(\mu(K_{i_j+1})+1)^{1/p}}.
\end{equation*}
Then $p^*\in P_{i_j+1}\subseteq P$ and we have that
\begin{equation*}
  \sup_{x\in X}\,\lvert g(x)-p^*(x)\rvert
  <\frac{\epsilon}{2(\mu(K_{i_j+1})+1)^{1/p}}
\end{equation*}
since the support $K$ of $g$ is contained in $K_{i_j+1}$ and the support of $p^*$ is
contained in $K_{i_j+1}$. Hence,
\begin{equation*}
  \lvert\lvert g-p^*\rvert\rvert_p^p
  =\int_{K_{i_j+1}}\!\lvert g-p^*\rvert^p\,d\mu
  \leq\frac{\epsilon^p}{2^p(\mu(K_{i_j+1})+1)}\mu(K_{i_j+1})
  <\Bigl(\frac{\epsilon}{2}\Bigr)^p,
\end{equation*}
so $\lvert\lvert g-p^*\rvert\rvert_p<\epsilon/2$ and hence
$\lvert\lvert f-p^*\rvert\rvert_p<\epsilon$.
