Probability-density approximation of continuous tests and topological means

For bounded continuous ${\psi }_1,\dots ,{\psi }_k$, let $D\subseteq {\mathbb{C}}^k$ consist of their integral vectors over $\mathcal{P}$, and let $v=(m({\psi }_j){)}_j$. The set $D$ is convex. If $v\mathrm{\notin }\bar{D}$, choose $r>0$ so that the open ball $B(v,r)$ is disjoint from the nonempty closed convex set $\bar{D}$. By [F3], after reversing its sign, a nonzero real-linear functional $\mathrm{\ell }$ satisfies $\mathrm{\ell }(c)<\mathrm{\ell }(u)$ for every $c\in \bar{D}$ and $u\in B(v,r)$. Choose a unit vector $w$ with $\mathrm{\ell }(w)>0$. Testing at $u=v-rw\mathrm{/}2$ gives ${\sup }_{d\in D}\mathrm{\ell }(d)\le \mathrm{\ell }(v)-r\mathrm{\ell }(w)\mathrm{/}2<\mathrm{\ell }(v)$. Write $h(x)=\mathrm{\ell }(({\psi }_j(x){)}_j)$, a real bounded continuous function. Complex linearity and positivity make $m(\mathrm{Re}\psi )=\mathrm{Re}m(\psi )$ and similarly for imaginary parts, so $\mathrm{\ell }(v)=m(h)\le {\sup }_{G}h$ by [F1]. But [F2] gives ${\sup }_{d\in D}\mathrm{\ell }(d)={\sup }_{f\in \mathcal{P}}\int fh={\sup }_{G}h$, a contradiction. Thus $v\in \bar{D}$, proving simultaneous continuous-test approximation by a density. The empty test family has a witness given by a normalized indicator of a relatively compact nonempty open set by [F4].

If $a=0$ or $b=0$, the adjoint identity below has both sides zero; assume otherwise, so both supports are nonempty. For $a,b\in C_c(G)$ define $a^{\mathrm{♯}}(u)={\mathrm{\Delta }}_G(u^{-1})a(u^{-1})$. For bounded continuous $\phi$, the compact-kernel formula and [F7] interchange the integrals of $a(y)b(z)\phi (yz)$; left invariance and inversion [F6] then give $\int (a\ast b)\phi =\int b(a^{\mathrm{♯}}\ast \phi )$. This also holds for any bounded Borel $\phi$. Indeed put $K=\mathrm{supp}a$, $L=\mathrm{supp}b$, $C=KL$, and choose $\psi \in C_c(G)$ with $\mathrm{\parallel }\psi -\phi {\mathbf{1}}_C{\mathrm{\parallel }}_1<\eta$ by [F4]. The left integral changes by at most $\mathrm{\parallel }a\ast b{\mathrm{\parallel }}_{\sup }\eta$. For $z\in L$, inversion and right translation give ${\int }_{K^{-1}}\mathrm{\mid }\phi -\psi \mathrm{\mid }(u^{-1}z)d\mu (u)\le M_K{M}_L\eta$, where $M_K={\sup }_K{\mathrm{\Delta }}_G(t^{-1})$ and $M_L={\sup }_L{\mathrm{\Delta }}_G(z^{-1})$ are finite by [F6]; here $tz\in KL=C$ for $t\in K$. The right integral changes by at most $\mathrm{\parallel }b{\mathrm{\parallel }}_1\mathrm{\parallel }{a}^{\mathrm{♯}}{\mathrm{\parallel }}_{\sup }{M}_K{M}_L\eta$. Letting $\eta \downarrow 0$ proves the adjoint identity for bounded Borel tests without invoking product measurability of arbitrary Borel functions.

These witnesses can be chosen in $\mathcal{P}\cap C_c(G)$. For a witness $b\in \mathcal{P}$, choose $u_n\in C_c(G)$ with $u_n\to b$ in $L^1$ by [F4]. Then $\mathrm{\mid }{u}_n\mathrm{\mid }\to b$ in $L^1$ and $\int \mathrm{\mid }{u}_n\mathrm{\mid }\to 1$; for large $n$, $b_n=\mathrm{\mid }{u}_n\mathrm{\mid }\mathrm{/}\int \mathrm{\mid }{u}_n\mathrm{\mid }$ lies in $\mathcal{P}\cap C_c(G)$ and tends to $b$ in $L^1$. The finite bounded tests preserve the desired inequalities with any initial smaller error margin. Index all such witnesses by triples $(F,\eta ,b)$, with finite continuous test set $F$, tolerance $\eta >0$ and $b\in \mathcal{P}\cap C_c(G)$ meeting it, ordered by increasing $F$ and decreasing $\eta$. Step 1.1 makes this a nonempty directed preorder. Its third-coordinate net $b_i$ satisfies $\int b_i\psi \to m(\psi )$ for every bounded continuous $\psi$, without a global witness choice. Fix also $p\in \mathcal{P}\cap C_c(G)$.

Now suppose $m$ is topologically invariant. Set $g_i=p^{\mathrm{♯}}\ast b_i$ for the Cc probability witnesses of step 2.1. By [F6], $p^{\mathrm{♯}}\ge 0$, $\mathrm{\parallel }{p}^{\mathrm{♯}}{\mathrm{\parallel }}_1=1$ and $(p^{\mathrm{♯}}{)}^{\mathrm{♯}}=p$, so $g_i\in \mathcal{P}$ by [F8]. For a global $L^{\mathrm{\infty }}$ class $\phi$, choose a bounded Borel representative by modifying a global null set. Step 1.2 gives $\int g_i\phi =\int b_i(p\ast \phi )\to m(p\ast \phi )=m(\phi )$, because [F5] makes $p\ast \phi$ bounded continuous and $m$ is topological. Thus the probability integrals converge weak-star on every class; in particular any finite family is approximated simultaneously. No locally-null identification was used. Together with steps 1.1 and 2.1 this proves both claims.