Do we know the approximate form of measurement operators for a realistic measuring apparatus?

For instance, consider a cloud/bubble/spark chamber. What operator $\hat X$ can be considered to measure the position from this? I assume it would have some properties such as being a bounded operator (since it won't measure any position outside of the apparatus itself), and that we won't measure $x$ itself but rather some mollified version of it, so that the operator might be something of the form

$$\langle \psi, \hat X \psi \rangle = \int_{\Omega_{\text{App.}}} \psi^*(x) f(x) \psi(x) dx$$

Do we know approximately what such an operator might be for various types of measuring apparatus, is there some standard treatment on the topic?


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