Lipschitz observable bias under bounded law shift

Let $(\mathcal{Z},\mathrm{\Sigma })$ be a measurable space and let $\mu$ be a probability measure on it, meaning a countably additive measure on $\mathrm{\Sigma }$ with $\mu (\mathcal{Z})=1$. Let $X$ be a real normed vector space and give it the norm-Borel $\sigma$-algebra generated by its norm-open sets. Let $S,\tilde{S}:\mathcal{Z}\to X$ be measurable maps. Let $J:X\to \mathbb{R}$ be $L$-Lipschitz, meaning $\mathrm{\mid }J(x)-J(y)\mathrm{\mid }\le L\mathrm{\parallel }x-y{\mathrm{\parallel }}_X$ for all $x,y\in X$, with $L\ge 0$. Suppose

${\int }_{\mathcal{Z}}(\mathrm{\parallel }S{\mathrm{\parallel }}_X^2+\mathrm{\parallel }\tilde{S}{\mathrm{\parallel }}_X^2)d\mu <\mathrm{\infty },{\int }_{\mathcal{Z}}\mathrm{\parallel }S-\tilde{S}{\mathrm{\parallel }}_X^{2}d\mu \le {\epsilon }^2$

for some finite $\epsilon \ge 0$. For a measurable scalar function $h$, write $\mathrm{\parallel }h{\mathrm{\parallel }}_{L^2(\mu )}=({\int }_{\mathcal{Z}}\mathrm{\mid }h{\mathrm{\mid }}^{2}d\mu {)}^{1\mathrm{/}2}$. Let $g:\mathcal{Z}\to [0,\mathrm{\infty })$ be measurable with $\int gd\mu =1$ and $g\le C$ $\mu$-almost everywhere for a finite $C\ge 1$, and define the target probability law by $\nu (E)={\int }_{E}gd\mu$ for $E\in \mathrm{\Sigma }$, with target expectation ${\mathbb{E}}_{\nu }[H]=\int Hgd\mu$ whenever the integral exists. Then the measurable scalar observables $J\circ S$ and $J\circ \tilde{S}$ are $\nu$-integrable and

$\mid {\mathbb{E}}_{\nu }[J(S)]-{\mathbb{E}}_{\nu }[J(\tilde{S})]\mid \le L\sqrt{C}\epsilon \mathrm{.}$

The estimate assumes a population mean-square error under $\mu$ and a bounded target density $g$; a finite test sample by itself establishes neither assumption. The state space may represent fluid fields when that measurable state model is supplied. For a physical application, both observables must use the same declared scalar unit; this mathematical proposition assigns no physical units.

Sheldon Axler, Measure, Integration & Real Analysis, Theorem 7.9 in §7A.