I don't have enough reputation to comment on Chiral Anomaly's answer so I am putting this here.

I think it is important to emphasise that nothing in this answer is specific to quantum field theory, and that this collection of operators $D(R)$ is a special case of the approach a quantum information theorist would use to answer this question.

A more general approach to quantum measurement, which is very widely used in the quantum information literature is the idea of a positive operator valued measure, which assigns positive operators to the measurable sets of a measure space and obeys some axioms which look a lot like the axioms of a probability measure (POVMs in particular satisfy an axiom of countable additivity so the operator assigned to the union of two disjoint sets is just the sum of the operator assigned to each of the sets individually).

This paper https://arxiv.org/abs/1604.00566, for example employs exactly the position observable mentioned in Chiral Anomaly's answer.

I mention this mainly because although in this case the natural position observable happens to be a projection valued POVM, in general the values of our POVM are free to be any positive operators which sum to the identity.