A topological invariant mean yields norm-approximately invariant densities

Choose an open relatively compact neighborhood $V$ of $e$. By [F2, F3], $0<\mu (V)<\mathrm{\infty }$, and $p_0:=\mu (V{)}^{-1}{\mathbf{1}}_V$ belongs to $\mathcal{P}$; this also proves $\mathcal{P}$ is nonempty.

Let ${\psi }_1,\dots ,{\psi }_n$ be bounded continuous complex functions and put $z(x)=({\psi }_1(x),\dots ,{\psi }_n(x))\in {\mathbb{C}}^n$, viewed as a real vector space with its maximum norm. The vector $v=(\tilde{m}({\psi }_1),\dots ,\tilde{m}({\psi }_n))$ lies in $C:=\stackrel{}{\mathrm{co}}(z(G))$: otherwise a ball $B(v,r)$ and $C$ are disjoint convex sets, so [F8] strictly separates them by a nonzero real-linear functional $\mathrm{\ell }$. Choose a unit vector $w$ with $\mathrm{\ell }(w)>0$; applying separation to $v+rw\mathrm{/}2\in B(v,r)$ gives $\mathrm{\ell }(v)<{\inf }_{c\in C}\mathrm{\ell }(c)={\inf }_{x\in G}\mathrm{\ell }(z(x))$. Now $h(x):=\mathrm{\ell }(z(x))$ is bounded continuous and real-valued, and $\mathrm{\ell }(v)=\tilde{m}(h)$. For real $u$, $u=u_{+}-u_{-}$ makes $\tilde{m}(u)$ real, so $\tilde{m}(\mathrm{Re}\psi )=\mathrm{Re}\tilde{m}(\psi )$ and similarly for imaginary parts; positivity and normalization give $\tilde{m}(h)\ge {\inf }_{G}h$, a contradiction.

We first show that every bounded complex-linear functional $\mathrm{\Lambda }$ on $L^1(G)$ is represented by a bounded Borel function. Let $H={\bigcup }_n{U}^n$ be as in [F9], choose one representative $t_C$ for each clopen left coset $C=t_{C}H$ using [A1], and put $K_{C,n}=t_C{U}^n$. Each $C$ is the increasing union of compact sets $K_{C,n}$ of finite Haar measure, hence its restricted Haar measure is sigma-finite. Since $C$ is clopen, it is an LCH subspace and its restricted measure is Radon: Borel subsets of $C$ are Borel in $G$, and open subsets of $C$ are open in $G$.

For each coset $C$, restrict $\mathrm{\Lambda }$ to complex $L^1$ functions supported in $C$. On real functions its real and imaginary parts are bounded real-linear functionals of norm at most $\parallel \mathrm{\Lambda }\mathrm{\parallel }$; [F11] represents them by real $a_C,b_C\in L^{\mathrm{\infty }}(C)$ with $\parallel a_C{\parallel }_{\mathrm{\infty }},\parallel b_C{\parallel }_{\mathrm{\infty }}\le \parallel \mathrm{\Lambda }\parallel$. Thus ${\varphi }_C:=a_C+i{b}_C$ represents $\mathrm{\Lambda }$ on complex functions supported in $C$ and $\mathrm{\mid }{\varphi }_C\mathrm{\mid }\le 2\parallel \mathrm{\Lambda }\parallel$ almost everywhere. Choose Borel representatives and set them to zero on their exceptional null sets.

Put $R=2\parallel \mathrm{\Lambda }\parallel$. For each $C,n$, the bounded function ${\varphi }_C{\mathbf{1}}_{K_{C,n}}$ is in $L^1(C)$; by [F13] choose $q_{C,n}\in C_c(C)$ within $2^{-n-2}$ in $L^1$. Radially clip $q_{C,n}$ to the closed disk of radius $R$, obtaining $h_{C,n}\in C_c(C)$ with $\parallel h_{C,n}-{\varphi }_C{\mathbf{1}}_{K_{C,n}}{\parallel }_1<2^{-n-1}$: pointwise, if $\mathrm{\mid }{q}_{C,n}\mathrm{\mid }>R$ then $\mathrm{\mid }{q}_{C,n}-h_{C,n}\mathrm{\mid }=\mathrm{\mid }{q}_{C,n}\mathrm{\mid }-R\le \mathrm{\mid }{q}_{C,n}-{\varphi }_C{\mathbf{1}}_{K_{C,n}}\mathrm{\mid }$, and otherwise clipping does nothing, so the triangle inequality gives the stated factor two.

If $f=0$ or $v=0$, the adjoint identity below has both sides zero. Otherwise their compact supports are nonempty. For $f,v\in C_c(G)$ and bounded Borel $\phi$, define $f^{\mathrm{♯}}(u):={\mathrm{\Delta }}_G(u^{-1})f(u^{-1})$. The adjoint identity ${\int }_G(f\ast v)\phi ={\int }_{G}v(f^{\mathrm{♯}}\ast \phi )$ holds first for bounded continuous $\phi$: apply [F22] to the compactly supported continuous kernel $(y,z)\mapsto f(y)v(z)\phi (yz)$, use left invariance to set $x=yz$, interchange the compact Radon integrals, and use inversion [F15] in ${\int }_{y}f(y)\phi (yz)$.

If the finite test family is empty, take $b:=p_0$ from step 1.1. Otherwise, given $\epsilon >0$, choose a finite convex combination ${\sum }_{k=1}^r{t}_{k}z(x_k)$ within $\epsilon \mathrm{/}2$ of $v$. Continuity of the finite family gives, for each $k$, an open neighborhood $O_k$ of $x_k$ on which $\mathrm{\mid }{\psi }_j(y)-{\psi }_j(x_k)\mathrm{\mid }<\epsilon \mathrm{/}2$ for every $j$; shrink it to a relatively compact open $V_k\subseteq O_k$. Then $p_k:=\mu (V_k{)}^{-1}{\mathbf{1}}_{V_k}\in \mathcal{P}$ by [F3], and $b:={\sum }_k{t}_k{p}_k\in \mathcal{P}$ by [F1] satisfies $\mathrm{\mid }\int b{\psi }_{j}d\mu -\tilde{m}({\psi }_j)\mathrm{\mid }<\epsilon$ for every $j$.

For fixed $C,m$ and every $n\ge m$, $K_{C,m}\subseteq K_{C,n}$, so ${\int }_{K_{C,m}}\mathrm{\mid }{h}_{C,n}-{\varphi }_C\mathrm{\mid }d\mu <2^{-n-1}$. By [F14], the sum of these errors is finite almost everywhere on $K_{C,m}$, hence $h_{C,n}\to {\varphi }_C$ almost everywhere there. Since $C={\bigcup }_m{K}_{C,m}$, convergence holds almost everywhere on each coset. Glue $h_{C,n}$ over the clopen cosets to a bounded continuous $H_n$ on $G$. Applying the convergence-set and pointwise-limit clauses of [F14] to the real and imaginary parts shows that the convergence set of $(H_n)$ is Borel and its limit there is Borel; set it to zero elsewhere to obtain a bounded Borel function $\varphi$ agreeing almost everywhere with each ${\varphi }_C$ on that coset. This uses only countable unions of exceptional null sets inside each individual coset.

For general bounded Borel $\phi$, let $K=\mathrm{supp}f$, $L=\mathrm{supp}v$, and $C=KL$, which is compact; the pointwise Cc convolution $f\ast v$ vanishes off $C$. Since $f^{\mathrm{♯}}\in L^1$ and $\phi$ is bounded Borel, [F21] makes $f^{\mathrm{♯}}\ast \phi$ bounded continuous, so the right pairing is defined. Choose $\psi \in C_c(G)$ with $\parallel \psi -\phi {\mathbf{1}}_C{\parallel }_1<\eta$ by [A1, F13]. The left pairing changes by at most $\parallel f\ast v{\parallel }_{\sup }\eta$. For fixed $z\in L$, inversion and right translation give ${\int }_{K^{-1}}\mathrm{\mid }\phi -\psi \mathrm{\mid }(u^{-1}z)d\mu (u)={\int }_K\mathrm{\mid }\phi -\psi \mathrm{\mid }(tz){\mathrm{\Delta }}_G(t^{-1})d\mu (t)\le M_K{M}_L\eta$, where $M_K={\sup }_{t\in K}{\mathrm{\Delta }}_G(t^{-1})$ and $M_L={\sup }_{z\in L}{\mathrm{\Delta }}_G(z^{-1})$ are finite by [F17]. Since $f^{\mathrm{♯}}$ is bounded and supported in $K^{-1}$, the right pairing changes by at most $\parallel v{\parallel }_1\parallel f^{\mathrm{♯}}{\parallel }_{\mathrm{\infty }}{M}_K{M}_L\eta$. Letting $\eta \downarrow 0$ proves the identity for Cc $f,v$ and arbitrary $\phi$, without assuming a Borel product function is measurable for the product sigma-algebra.

Index by triples $(F,\epsilon ,b)$ where $F$ is a finite set of bounded continuous functions, $\epsilon >0$, $b\in \mathcal{P}$, and $\mathrm{\mid }\int b\psi -\tilde{m}(\psi )\mathrm{\mid }<\epsilon$ for all $\psi \in F$; order by inclusion of $F$ and decreasing $\epsilon$. Steps 1.1 and 2.1 make this a nonempty directed preorder, since a witness for the union of two finite test sets and the smaller tolerance gives a common upper bound. The third-coordinate net $(b_i)$ therefore satisfies $\int b_i\psi d\mu \to \tilde{m}(\psi )$ for every bounded continuous $\psi$, without a global choice function.

A compact set meets only finitely many open cosets $C$, since these cosets form an ambient open cover and [F27] supplies a finite subcover. Hence for every $u\in C_c(G)$, step 1.4 and the cosetwise agreement in step 2.2 give $\mathrm{\Lambda }(u)={\int }_{G}u\varphi d\mu$. By [A1, F13], $C_c(G)$ is dense in $L^1(G)$; both sides are bounded functionals, with the integral norm at most $2\parallel \mathrm{\Lambda }\parallel$, so equality extends to every $u\in L^1(G)$. Thus the full bounded dual of $L^1(G)$ is represented by bounded Borel functions, without claiming that an uncountable union of null sets is null.

For arbitrary $f,v\in L^1(G)$, approximate them in $L^1$ by Cc sequences. The class convolution bound [F19] makes $f_n\ast v_n\to f\ast v$ in $L^1$; the inversion formula gives $\parallel f_n^{\mathrm{♯}}-f^{\mathrm{♯}}{\parallel }_1=\parallel f_n-f{\parallel }_1$, and the smoothing bound [F21] gives $f_n^{\mathrm{♯}}\ast \phi \to f^{\mathrm{♯}}\ast \phi$ uniformly. Passing to the limit in step 2.3 proves ${\int }_G(f\ast v)\phi ={\int }_{G}v(f^{\mathrm{♯}}\ast \phi )$ for all $f,v\in L^1(G)$ and $\phi \in L^{\mathrm{\infty }}(G)$.

Fix $p:=p_0\in \mathcal{P}$ from step 1.1 and set $g_i:=p^{\mathrm{♯}}\ast b_i$. Formula [F15] gives $\parallel p^{\mathrm{♯}}{\parallel }_1=\parallel p{\parallel }_1=1$, preserves nonnegativity, and yields $(p^{\mathrm{♯}}{)}^{\mathrm{♯}}=p$; the earlier convolution-closure proof [F20] gives $g_i\in \mathcal{P}$. For every $\phi \in L^{\mathrm{\infty }}(G)$, step 3.3 yields $\int g_i\phi =\int b_i(p\ast \phi )\to \tilde{m}(p\ast \phi )=\tilde{m}(\phi )$, since $p\ast \phi$ is bounded continuous by [F21] and $\tilde{m}$ is topologically invariant. Thus the density functionals $g_i$ converge weak-star to $\tilde{m}$ on all of $L^{\mathrm{\infty }}(G)$, although the net $b_i$ was only chosen to approximate on continuous tests.

Fix $f\in \mathcal{P}$. Formula [F15] shows that $f^{\mathrm{♯}}\ge 0$ and $\mathrm{\parallel }{f}^{\mathrm{♯}}{\mathrm{\parallel }}_1=\mathrm{\parallel }f{\mathrm{\parallel }}_1=1$, so $f^{\mathrm{♯}}\in \mathcal{P}$. For every $\phi \in L^{\mathrm{\infty }}(G)$, step 3.3 gives $\int (f\ast g_i)\phi =\int g_i(f^{\mathrm{♯}}\ast \phi )\to \tilde{m}(f^{\mathrm{♯}}\ast \phi )=\tilde{m}(\phi )$ by topological invariance, while step 4.1 gives $\int g_i\phi \to \tilde{m}(\phi )$. Hence $\int (f\ast g_i-g_i)\phi \to 0$. By the full dual representation in step 3.2, every bounded functional on $L^1(G)$ is one of these pairings; therefore $f\ast g_i-g_i\rightharpoonup 0$ weakly in $L^1(G)$.

Let $F=\{f_1,\dots ,f_r\}\subseteq \mathcal{P}$ be finite and nonempty. The tuple $(f_k\ast g_i-g_i{)}_{k=1}^r$ converges weakly to zero in $E=(L^1(G){)}^r$ with its maximum norm: each coordinate converges weakly by step 5.1, and every bounded functional on this finite product is the sum of its coordinate restrictions. Its range over $i$ lies in the convex set $D=\{(f_k\ast g-g{)}_{k=1}^r:g\in \mathcal{P}\}$, since the map is linear in $g$ and $\mathcal{P}$ is convex. Thus zero is in the weak closure of $D$; [F25] puts it in the norm closure, so for every $\epsilon >0$ some $g\in \mathcal{P}$ satisfies ${\max }_k\parallel f_k\ast g-g{\parallel }_1<\epsilon$. The empty $F$ has any witness from step 1.1.

Index by triples $(F,\epsilon ,g)$ with finite $F\subseteq \mathcal{P}$, $\epsilon >0$, $g\in \mathcal{P}$, and $\parallel f\ast g-g{\parallel }_1<\epsilon$ for all $f\in F$, ordered by inclusion of $F$ and decreasing tolerance. Step 6.1 makes this a nonempty directed preorder, so its third-coordinate net $(g_j)$ lies in $\mathcal{P}$ and satisfies $\parallel f\ast g_j-g_j{\parallel }_1\to 0$ for each $f\in \mathcal{P}$, without a global choice function.

Let $C\subseteq \mathcal{P}$ be norm-compact and $\delta >0$. If $C=\mathrm{\varnothing }$ the estimate is vacuous; otherwise choose a finite $\delta \mathrm{/}3$-net $f_1,\dots ,f_r\in C$ and take an index after these tests with tolerance $\delta \mathrm{/}3$. For every later $j$, choose $f_k$ with $\parallel f-f_k{\parallel }_1<\delta \mathrm{/}3$; since $g_j\in \mathcal{P}$, [F19] gives $\parallel f\ast g_j-g_j{\parallel }_1\le \parallel (f-f_k)\ast g_j{\parallel }_1+\parallel f_k\ast g_j-g_j{\parallel }_1<2\delta \mathrm{/}3<\delta$. This proves uniform convergence on $C$ and completes the lemma.