Operators commuting with a generating family of multiplications

If $\mu (X)=0$, then $H=0$, $T=M_0$, and $m=0$ is constant, so both conclusions hold. Hence assume $\mu (X)>0$, which by $\sigma$-finiteness implies $H\mathrm{\ne }0$. For bounded real $h$, let $R(h)=\{\lambda \in \mathbb{R}:\mu (\{\mathrm{\mid }h-\lambda \mathrm{\mid }<ϵ\})>0\text{ for every }ϵ>0\}$. Its complement is a union of countably many rational intervals with null preimages, so $h\in R(h)$ a.e.; consequently $R(h)$ is nonempty, closed and bounded. The operator $M_h$ is bounded self-adjoint. If $\lambda \mathrm{\notin }R(h)$ is real, then $\mathrm{\mid }h-\lambda \mathrm{\mid }\ge ϵ$ a.e. for some $ϵ>0$, giving the bounded inverse $M_{(h-\lambda {)}^{-1}}$ of $M_h-\lambda I$; for nonreal $\lambda$, use $\mathrm{\mid }h-\lambda \mathrm{\mid }\ge \mathrm{\mid }\mathrm{Im}\lambda \mathrm{\mid }$. If $\lambda \in R(h)$, $\sigma$-finiteness supplies a measurable $E\subseteq \{\mathrm{\mid }h-\lambda \mathrm{\mid }<ϵ\}$ with $0<\mu (E)<\mathrm{\infty }$. The unit vector ${\mathbf{1}}_E\mathrm{/}\sqrt{\mu (E)}$ has image under $M_h-\lambda I$ of norm at most $ϵ$, ruling out a bounded inverse. Thus $\sigma (M_h)=R(h)$. On Borel subsets of $R(h)$ define $F(B)=M_{{\mathbf{1}}_{h^{-1}(B)}}$. These are orthogonal projections with $F(R(h))=I$, and disjoint unions give strong countable additivity by dominated convergence applied to $\mathrm{\mid }v{\mathrm{\mid }}^2$ for each $v\in H$. The scalar measures $⟨F(B)v,v⟩={\int }_{h^{-1}(B)}\mathrm{\mid }v{\mathrm{\mid }}^{2}d\mu$ are finite, hence regular on the compact subset $R(h)$ of $\mathbb{R}$ by [F5]. Integrating simple functions and then uniform approximants gives ${\mathrm{\Phi }}_F(f)=M_{f\circ h}$ for continuous $f$, in particular ${\mathrm{\Phi }}_F(z)=M_h$. Uniqueness in [F3] identifies $F$ as the spectral PVM of $M_h$, so ${\mathbf{1}}_B(M_h)=M_{{\mathbf{1}}_{h^{-1}(B)}}$ for every Borel $B$; for $B\subseteq \mathbb{R}$ use $B\cap R(h)$.

If $N$ is bounded normal and $S\in \mathcal{B}(H)$ commutes with $N$ and $N^{\ast }$, then $S$ commutes with $f(N)$ for every bounded Borel function $f$ on $\sigma (N)$; in particular $S$ commutes with every spectral projection ${\mathbf{1}}_B(N)=E(B)$. This is the commutation clause of the bounded Borel calculus [F3].

Let $W:=\{g\in L^{\mathrm{\infty }}(\mu ):T{M}_g=M_{g}T\}$ be the set of bounded measurable functions whose multiplication commutes with $T$. Then $W$ contains the constants, is a complex vector space, is closed under products by [F2], and is closed under bounded pointwise almost-everywhere convergence: if $g_n\in W$, $\mathrm{\parallel }{g}_n{\mathrm{\parallel }}_{\mathrm{\infty }}\le C$ and $g_n\to g$ pointwise a.e., then for $\phi \in H$ the dominated convergence theorem [F4] gives $g_n\phi \to g\phi$ and $g_n(T\phi )\to g(T\phi )$ in $L^2$, while $T(g_n\phi )=g_{n}T\phi$; passing to the limit gives $T(g\phi )=g(T\phi )$, that is, $g\in W$.

If $T$ commutes with $M_h$ for a bounded real measurable $h$, then $T$ commutes with $M_{{\mathbf{1}}_{h^{-1}(B)}}$ for every Borel $B\subseteq \sigma (M_h)$: since $M_h$ is self-adjoint, $T$ commutes with $M_h$ and $M_h^{\ast }=M_h$, so step 1.2 makes $T$ commute with ${\mathbf{1}}_B(M_h)$, which equals $M_{{\mathbf{1}}_{h^{-1}(B)}}$ by step 1.1.

The set $\mathcal{D}:=\{B\in \mathrm{\Sigma }:{\mathbf{1}}_B\in W\}$ contains the algebra generated by the cylinder sets $h^{-1}(B^{\mathrm{\prime }})$ with $h\in \mathcal{F}$ and $B^{\mathrm{\prime }}$ Borel: each such indicator lies in $W$ by step 2.1 (completing $\sigma (\mathcal{F})$ to $\mathrm{\Sigma }$ only affects null sets, on which indicators differ by $W$-elements), and $W$ is closed under finite linear combinations and products by step 1.3, so finite unions and intersections of cylinders have indicators in $W$. Moreover $\mathcal{D}$ is closed under complements (as $1-{\mathbf{1}}_B\in W$) and under increasing countable unions (as indicators converge boundedly pointwise), so it is a monotone class; by [F6] it contains $\sigma (\mathcal{F})$ and hence, after completing by null sets, all of $\mathrm{\Sigma }$. Since every bounded $\mathrm{\Sigma }$-measurable function is a bounded pointwise limit of simple functions, step 1.3 shows $W=L^{\mathrm{\infty }}(\mu )$: $T$ commutes with $M_g$ for every bounded measurable $g$.

Consequently $T=M_m$ for a bounded measurable $m$. Choose measurable sets $E_n$ of finite measure with $E_n\uparrow X$ up to a null set (possible by $\sigma$-finiteness), and put $u_n:=T{\mathbf{1}}_{E_n}\in H$, $m_n:=u_n{\mathrm{\mid }}_{E_n}$. For every bounded measurable $f$ supported in $E_n$ one has $Tf=T(M_f{\mathbf{1}}_{E_n})=M_{f}T{\mathbf{1}}_{E_n}=m_{n}f$ by step 3.1, so in particular $m_n{\mathbf{1}}_B=T{\mathbf{1}}_B$ for every measurable $B\subseteq E_n$ and $\mathrm{\parallel }{m}_n{\mathbf{1}}_B{\mathrm{\parallel }}_2\le \mathrm{\parallel }T\mathrm{\parallel }\mathrm{\parallel }{\mathbf{1}}_B{\mathrm{\parallel }}_2=\mathrm{\parallel }T\mathrm{\parallel }\mu (B{)}^{1\mathrm{/}2}$. Applying this to $B=\{\mathrm{\mid }{m}_n\mathrm{\mid }>\mathrm{\parallel }T\mathrm{\parallel }+ϵ\}\cap E_n$ gives $(\mathrm{\parallel }T\mathrm{\parallel }+ϵ)\mu (B{)}^{1\mathrm{/}2}\le \mathrm{\parallel }T\mathrm{\parallel }\mu (B{)}^{1\mathrm{/}2}$, hence $\mu (B)=0$ and $\mathrm{\parallel }{m}_n{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }T\mathrm{\parallel }$ on $E_n$; the functions $m_n$ therefore glue (they agree a.e. on overlaps by the same computation with $B\subseteq E_n\cap E_k$) to a bounded measurable $m$. For bounded measurable $f$ supported in one $E_n$, the already established identity gives $Tf=m_{n}f=M_{m}f$. These functions are dense in $H$: for arbitrary $f\in H$, the bounded functions $f{\mathbf{1}}_{E_n}{\mathbf{1}}_{\{\mathrm{\mid }f\mathrm{\mid }\le n\}}$ converge to $f$ in $L^2$ by [F4]. Since $T$ and $M_m$ are bounded, $T=M_m$.

For the ergodic clause, observe first that if $S$ is measure preserving with induced unitary $U_{S}f=f\circ S^{-1}$ and $T=M_m$ commutes with $U_S$, then $m=m\circ S^{-1}$ a.e.: $U_S{M}_m{U}_S^{-1}=M_{m\circ S^{-1}}$ because $(U_S{M}_m{U}_S^{-1})f=(m\cdot (f\circ S))\circ S^{-1}=(m\circ S^{-1})f$, so $M_m=U_S{M}_m{U}_S^{-1}$ and multiplication by two functions agrees only if the functions agree a.e. [F2]. Hence every rational level set $A_q:=\{\mathrm{Re}m>q\}$ satisfies $\mu (A_q\mathrm{△}{S}^{-1}{A}_q)=0$ for every $S$ in the family, so each $A_q$ is invariant modulo null sets; ergodicity gives $\mu (A_q)=0$ or $\mu (X\setminus A_q)=0$. The set $\{q\in \mathbb{Q}:\mu (A_q)=0\}$ is a final segment with finite infimum $a$, because $m$ is essentially bounded. On the conull set where all rational level sets are decided, every rational $q<a$ satisfies $\mathrm{Re}m>q$ and every rational $q>a$ satisfies $\mathrm{Re}m\le q$, so $\mathrm{Re}m=a$, hence constant a.e.; the same argument applied to $\mathrm{Im}m$ makes $m$ constant a.e.

The Axiom of Choice is inherited from the spectral, PVM-integral and measure-regularity suppliers of [F1]–[F6]; the commutant, exhaustion and ergodicity computations add no further choice (The Axiom of Choice).