An additive-noise POVM, not an apparatus claim

Consider one represented self-adjoint observable ${\hat{O}}_{ϵ}$. Let ${\nu }_{ϵ}$ be a classical probability law of mean zero and finite variance ${{\sigma }_{ϵ}}^2$, independent of the system. Define the POVM

as a weak operator integral. Positivity and normalization follow from the corresponding properties of the spectral measure and ${\nu }_{ϵ}$; countable additivity follows by monotone convergence.

Its outcome law is the convolution ${\mu }_{ϵ}\ast {\nu }_{ϵ}$, equivalently the law of $Y_{ϵ}=X_{ϵ}+{\eta }_{ϵ}$ for independent variables with those laws.