measurable-spectral-bijection

For every nonempty measurable space $M$, define the closed-sublevel family of $g:M\to\mathbb R$ by $E_g(t)=\{x:g(x)\le t\}$, and for any family $E:\mathbb R\to\mathcal P(M)$ define $c_E(x)=\inf\{t\in\mathbb R:x\in E(t)\}$, using the total real infimum, which is $0$ for an empty set or a set unbounded below. A measurable spectral family means that every $E(t)$ is measurable, $E(t)\subseteq E(s)$ whenever $t\le s$, $E(t)=\bigcap_{s>t}E(s)$ for every $t$, $\bigcap_{t\in\mathbb R}E(t)=\varnothing$, and $\bigcup_{t\in\mathbb R}E(t)=M$. The theorem asserts both that every measurable real-valued function $g$ has $E_g$ a measurable spectral family and $c_{E_g}=g$, and that every measurable spectral family $E$ has $c_E$ measurable, satisfies $E_{c_E}=E$, and has $\{t:x\in E(t)\}$ nonempty and bounded below for every $x\in M$. Measurability of real-valued functions uses the usual Borel measurable structure on $\mathbb R$; neither a bound on $g$ nor bounds uniform in $x$ on these cuts are assumed.