Condition for separability of $L^2_C(Z,nu)$ in Dixmier Von Neumann Algebras proof

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:

1. $mu(K)<infty$ for every compact set $Ksubseteq X$.
2. if $Einmathscr{B}$ and $mu(E)<infty$ then $$mu(E)=inf,{mu(V)colon Esubseteq V,text{ $V$ open}}.$$
3. if $Einmathscr{B}$ and $mu(E)<infty$ then $$mu(E)=sup,{mu(K)colon Ksubseteq E,text{ $K$ compact}}.$$ [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 $1leq p<infty$.

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 $C={Uin Dcolonbar{U}text{ is compact}}$. Then $C$ is also a countable base, for if $xin V$, 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 $xin Wsubseteqbar{W}subseteq V$. Then, for some $Uin D$, $xin Usubseteq Wsubseteq V$, so $bar{U}subseteqbar{W}$ and hence $bar{U}$ is compact, and therefore $Uin C$. 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}_1cupdotscupbar{U}_{j_i+1} subseteq K_{i+1}^circqquad(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 $K_isubseteq U_1cupdotscup U_{j_{i+1}}$, and let $K_{i+1}=bar{U}_1cupdotscupbar{U}_{j_{i+1}+1}$, which is compact. Then begin{equation*} K_isubseteq U_1cupdotscup U_{j_{i+1}}text{ open } subseteqbar{U}_1cupdotscupbar{U}_{j_{i+1}}cupbar{U}_{j_{i+1}+1}, end{equation*} so $K_isubseteq(bar{U}_1cupdotscupbar{U}_{j_{i+1}+1})^circ=K_{i+1}^circ$. If $xin X$ then $xin U_k$ for some $k$. Let $i$ be such that $j_igeq k$. Then $xin U_ksubseteqbar{U}_1cupdotscupbar{U}_{j_i}cupbar{U}_{j_i+1}=K_i$, 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 $s,tin S$. 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 $g_ncolon Stomathbb{C}$ by $g_n(s)=d(s,s_n)$. By KGT 4.8, $g_nin C(S)$, and we have that $0leq g_n(s)leq 1$ for all $sin S$. 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 $s,tin S$ with $sneq t$, 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 $$g_n(t)=d(t,s_n)geqepsilon-d(s,s_n)>epsilon/2>d(s,s_n)=g_n(s),$$ and hence $B$ separates points on $S$. If $sin S$, 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 $$check{p}(g)=sum_{lvertalpharvertleq N}q_alpha g^alpha,$$ where $alpha$ is a multi-index (see RFA 1.34 for the definition), $g=(g_{n_1},dots,g_{n_k})$, and $Re q_alpha,Im q_alphainmathbb{Q}$.

Let $epsilon>0$ be given along with a polynomial $pin B$, say $$p(g)=sum_{lvertalpharvertleq N}c_alpha g^alpha,$$ where $c_alpha$ is a complex number for each $alpha$ such that $lvertalpharvert<N$ and $g=(g_{n_1},dots,g_{n_k})$. Then $lvert g_{n_j}(s)rvertleq 1$ for all $sin S$ and $j=1,dots,k$, so for $sin S$ and $lvertalpharvertleq N$, begin{equation*} lvert g^alpha(s)rvert =lvert g_{n_1}^{alpha_1}(s)cdots g_{n_k}^{alpha_k}(s)rvertleq 1 qquad(sin S,,lvertalpharvertleq N). end{equation*} Then, if for each $alpha$ such that $lvertalpharvertleq N$, a rational complex number $q_alpha$ is chosen such that $$lvert c_alpha-q_alpharvert<frac{epsilon}{(N+1)^k},$$ then for all $sin S$, begin{equation*} lvert(p(g))(s)-(check{p}(g))(s)rvert leqsum_{lvertalpharvertleq N}lvert c_alpha-q_alpharvert,lvert g^alpha(s)rvert <sum_{lvertalpharvertleq 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 $check{p}incheck{B}_i$ to a function $p^*$ on $X$. Let $P_i={p^*coloncheck{p}incheck{B}_i}$ and put $$P=bigcup_{i=1}^infty P_i.$$ Then $P$ is countable and $Psubseteq L^p(mu)$ since $p^*$ is bounded and $mu(K_i)<infty$. Let $fin L^p(mu)$ 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 $gin C_c(X)$ such that $lvertlvert f-grvertrvert_p<epsilon/2$. Let $K$ be the support of $g$. Then $K$ is compact. If $xin Ksubseteq X$, then $xin K_isubseteq K_{i+1}^circ$ for some $i$. Therefore ${K_{i+1}^circcolon Kcap K_{i+1}^circneqvarnothing}$ is an open cover of $K$ and so $Ksubseteq K_{i_1+1}^circcupdotscup K_{i_j+1}^circsubseteq K_{i_j+1}$ for some $i_1<dots<i_j$. Hence $g|K_{i_j+1}in C(K_{i_j+1})$. Then there exists $check{p}incheck{B}_{i_j+1}$ such that begin{equation*} sup_{sin 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_{xin 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*} lvertlvert g-p^*rvertrvert_p^p =int_{K_{i_j+1}}!lvert g-p^*rvert^p,dmu leqfrac{epsilon^p}{2^p(mu(K_{i_j+1})+1)}mu(K_{i_j+1}) <Bigl(frac{epsilon}{2}Bigr)^p, end{equation*} so $lvertlvert g-p^*rvertrvert_p<epsilon/2$ and hence $lvertlvert f-p^*rvertrvert_p<epsilon$.