Steinhaus and Pettis: Borel homomorphisms of second-countable locally compact groups are continuous

Part (i): let $A$ be Borel with $0<\mu (A)<\mathrm{\infty }$ and put $f={\mathbf{1}}_A\in L^2(G)$. For $g\in G$, $L_{g}f={\mathbf{1}}_{gA}$, so $⟨L_{g}f,f⟩=\mu (gA\cap A)$; by [F2] this function of $g$ is continuous and equals $\mu (A)>0$ at $g=e$. Hence there is an open neighbourhood $W$ of $e$ with $\mu (gA\cap A)>0$, hence $gA\cap A\mathrm{\ne }\mathrm{\varnothing }$, for every $g\in W$; writing $g{a}_1=a_2$ with $a_1,a_2\in A$ gives $g=a_2{a}_1^{-1}\in A{A}^{-1}$. Thus $W\subseteq A{A}^{-1}$.

Part (ii), reduction: $\phi$ is a homomorphism, so for $g_0\in G$, $\phi (g)\phi (g_0{)}^{-1}=\phi (g{g}_0^{-1})$; since multiplication and inversion in $T$ are continuous by [F3], $\phi$ is continuous at every point as soon as it is continuous at $e$. It therefore suffices to show that ${\phi }^{-1}(U)$ is a neighbourhood of $e_G$ for every neighbourhood $U$ of $e_T$.

A second-countable space is Lindelöf, with the argument of [F5]: fixing a countable base, the basic open sets contained in some member of an open cover form a countable refinement, and Countable Choice (a consequence of AC) selects a cover member for each of them. We apply this to $\phi (G)$ with the subspace topology, which is second countable as a subspace of $T$.

The unitary group $U(K)$ of a separable Hilbert space is second countable in the strong operator topology: fix a finite or countable orthonormal basis $(e_n)$ padded to a sequence by [F7] and consider $\mathrm{\Psi }:U(K)\to K^{\mathbb{N}}$, $\mathrm{\Psi }(u)=(u{e}_n{)}_n$. It is injective (a unitary vanishing on a complete orthonormal system is zero) and continuous for the SOT by [F6]; conversely, if $u_i\to u$ in the initial topology of the coordinate maps $u\mapsto u{e}_n$, then for $\xi ={\sum }_n{c}_n{e}_n\in K$ and $\epsilon >0$ choose $N$ with ${\sum }_{n>N}\mathrm{\mid }{c}_n{\mathrm{\mid }}^2<{\epsilon }^2\mathrm{/}16$; then $\mathrm{\parallel }(u_i-u)\xi \mathrm{\parallel }\le {\sum }_{n\le N}\mathrm{\mid }{c}_n\mathrm{\mid }\mathrm{\parallel }(u_i-u)e_n\mathrm{\parallel }+2({\sum }_{n>N}\mathrm{\mid }{c}_n{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}<\epsilon \mathrm{/}2+\epsilon \mathrm{/}2$ for all $i$ beyond a suitable index, so $u_i\to u$ strongly. Thus the SOT on $U(K)$ is the initial topology of the countable family $(u\mapsto u{e}_n{)}_n$, making it homeomorphic to a subspace of the second-countable space $K^{\mathbb{N}}$, hence second countable.

$U(K)$ with the strong operator topology is a topological group: if $u_i\to u$ and $v_i\to v$ strongly, then $\mathrm{\parallel }(u_i{v}_i-uv)\xi \mathrm{\parallel }\le \mathrm{\parallel }{u}_i(v_i-v)\xi \mathrm{\parallel }+\mathrm{\parallel }(u_i-u)v\xi \mathrm{\parallel }\le \mathrm{\parallel }(v_i-v)\xi \mathrm{\parallel }+\mathrm{\parallel }(u_i-u)v\xi \mathrm{\parallel }\to 0$; and if $u_i\to u$ strongly with $u_i,u$ unitary, then $\mathrm{\parallel }(u_i^{-1}-u^{-1})\xi \mathrm{\parallel }=\mathrm{\parallel }{u}_i^{-1}(u-u_i)u^{-1}\xi \mathrm{\parallel }=\mathrm{\parallel }(u-u_i)u^{-1}\xi \mathrm{\parallel }\to 0$, so inversion is continuous.

Fix a neighbourhood $U$ of $e_T$ and choose a symmetric neighbourhood $V$ of $e_T$ with $V^2\subseteq U$, using continuity of multiplication at $(e_T,e_T)$ and symmetry of neighbourhoods ([F3]). The family $\{\phi (g)V\cap \phi (G):g\in G\}$ is an open cover of $\phi (G)$, because $\phi (g)\in \phi (g)V$; by [step 1.3] it has a countable subcover with centres $\phi (g_n)$, $n\in \mathbb{N}$, chosen with $g_n\in G$.

The preimage ${\phi }^{-1}(V)$ has positive measure: it is Borel by [F4], and if $\mu ({\phi }^{-1}(V))=0$, then for every $g\in G$ the left translate $g{\phi }^{-1}(V)$ is null by [F1] and the sets $g_n{\phi }^{-1}(V)$ cover $G$, since $\phi (g)\in \phi (g_n)V$ gives $\phi (g_n{)}^{-1}\phi (g)=\phi (g_n^{-1}g)\in V$, that is, $g_n^{-1}g\in {\phi }^{-1}(V)$ and $g\in g_n{\phi }^{-1}(V)$. A countable cover of the nonempty open set $G$ by null sets would give $\mu (G)=0$, contradicting positivity of $\mu$ on the nonempty open set $G$ in [F1].

Choose a Borel set $A\subseteq {\phi }^{-1}(V)$ with $0<\mu (A)<\mathrm{\infty }$: by [F8] and second countability, the members of a countable base with compact closure cover $G$, so $G={\bigcup }_m{K}_m$ with $K_m$ compact; if $\mu ({\phi }^{-1}(V)\cap K_m)=0$ for every $m$ then countable additivity would give $\mu ({\phi }^{-1}(V))=0$, contrary to [step 3.1], so some $A:={\phi }^{-1}(V)\cap K_m$ is Borel with $0<\mu (A)\le \mu (K_m)<\mathrm{\infty }$ by [F1].

By part (i), [step 1.1], the set $A{A}^{-1}$ contains an open neighbourhood of $e_G$, and $A{A}^{-1}\subseteq {\phi }^{-1}(V){\phi }^{-1}(V{)}^{-1}\subseteq {\phi }^{-1}(V{V}^{-1})={\phi }^{-1}(V^2)\subseteq {\phi }^{-1}(U)$, because $\phi$ is a homomorphism and $V$ is symmetric. Hence ${\phi }^{-1}(U)$ is a neighbourhood of $e_G$; by [step 1.2] $\phi$ is continuous. This proves (ii).

Let $\phi :G\to U(K)$ be a Borel homomorphism from the second-countable LCH group $G$ into $U(K)$ with the strong operator topology. By [step 1.4] $U(K)$ is second countable and by [step 1.5] it is a topological group, so part (ii) proved in [step 5.1] applies and $\phi$ is strongly continuous. Together with part (i) of [step 1.1] this proves every assertion of the statement.