On a time-resolved interpretation of the Husimi function

In this Letter, we interpret the Husimi function as the conditional probability density of continuously measuring a stream of constant position and momentum outcomes, indefinitely. This gives rise to an alternative definition that naturally extends to an arbitrary collection of self-adjoint operators without reference to coherent states. This definition recovers the Husimi distribution for a spin-half particle when monitoring the three Pauli matrices, as well as Born's rule for quantum measurement when monitoring commuting quantum observables. Ultimately, the proposed paradigm generates positive representations of quantum states as conditional densities, on both finite and infinite time classical experiments, as expectations of a fundamental operator, the Gaussian semigroup.