I've recently been dipping my toes deeper into the so-called "Wigner function" formalism for quantum theory, and what I am curious about is this: ostensibly, the Wigner function is the analogue of a classical probability density function, or pdf, over phase space - with the interesting property that it can take on negative values (so-called "negative probability"). Now, I have often heard that an integral of a Wigner over a finite region, i.e.
$$P[(x, p) \in R] = \iint_R W(x, p)\ dx\ dp$$
is "meaningless" presumably because you cannot simultaneously measure $x$ and $p$, due to the Heisenberg uncertainty limit. But that's actually not correct - in a naive sense, you just can't measure them to infinite precision. After all, the Heisenberg principle doesn't say you must have all or nothing, it says there is a limit, bounded by $\hbar$, and the Wigner function likewise cannot be localized to regions smaller than about $1\ \hbar$ in area in the phase space. Moreover, in the most-upvoted anwer here:
it is mentioned how that simultaneous measurement on $x$ and $p$ is in fact possible to arbitrary "precision" (in the Wigner formalism, presumably to arbitrarily below 1 $\hbar$ of phase space), you can still get "results", it's just that they will fail to reproduce, i.e. a second measurement immediately on the heels of the first will yield another pair of random values for $x$ and $p$, which makes perfect sense if you think of the probability distribution as becoming localized but not arbitrarily so, so that it retains non-trivial spread at all times.
Thus what I am wondering is: does the above integral, then, have any relation to such simultaneous measurements? Like, at the very least, if the area of the integration region $R$ is much, much larger than $\hbar$ (say $1000\ \hbar$ or more), and we're given a valid state that is widely spread in phase space, then does the integral converge to, say, the classical probability to find the particle with those position and momentum parameters? (Hence the suggestive notation above.) Moreover, at the small end of the scale, is the given $P$ in any way relatable to the probabilities involved in what is discussed above about simultaneous measurement on $x$ and $p$ (though note here it can be negative, suggesting further interpretation is required)?