 Threshold Choquet representation in L-infinity

Work with the real Lp space $L^{\infty }((0,1))$ for Lebesgue measure, so its predual $L^1((0,1))$ is separable. Choose a measurable representative $f$ of a unit-ball element with $-1\le f\le 1$ and putFor every $t$, $H_t$ has modulus one almost everywhere, so it is an extreme point of the real $L^{\infty }$ closed unit ball. The map $t\mapsto H_t$ is Borel for the weak-star topology: pairing with any $g\in L^1$ is measurable, and separability makes the weak-star unit ball metrizable. Push forward uniform probability measure on $[0,1]$ through this map.

For almost every $s$,Fubini's theorem then gives ${\int }_0^1\int g(s)H_t(s)dsdt=\int g(s)f(s)ds$ for every $g\in L^1$. Thus the resulting Borel probability measure is concentrated on the extreme points and has barycenter $f$.

For $f(s)=2s-1$, the profiles are the threshold steps $H_t(s)=21_{\{t\le s\}}-1$. Averaging these sign-valued extreme points produces the linear profile. This is an explicit Choquet theorem representation in a setting where the required metrizability is actually available.