Suppose $K$ is a non-Archimedian local field and suppose $\omega$ is a top-degree differential form on a compact $K$-analytic manifold $M$ of dimension $n$. It is possible to construct a Borel measure on $M$, denoted $\mu_{\omega}$, essentially by writing $$\mu_{\omega}(A) = \int_A|f|d\mu $$ locally. That is, on a coordinate patch $U$ on which $\omega = fdx_1\wedge...\wedge x_n$ we define it as such (we can make sense of this by integrating on polydisks and applying Caratheodory's extension theorem) with respect to the Haar measure $\mu$ on $K^n$. This doesn't depend on the choice of coordinates because of the standard change of variables formula.
One element of this construction is that an arbitrary Borel set $A$ can be written as a disjoint union of Borel sets which are contained in a coordinate patch on which we know how to integrate Borel sets. This leads me to some questions:
Does this extend to $K$-analytic manifolds which are not compact? I explicitly used compactness to show that I can cover $M$ with disjoint coordinate patches. (I did the obvious and used clopenness of the basis to iteratively transform a finite clopen cover into a finite disjoint clopen cover. This does not work if $M$ is not compact.) I do not believe it should, but I cannot come up with a counterexample.
I do not see why this construction fails for real manifolds. I have a (very strong) suspicion that this is defining a density on a manifold. In fact, I am nearly certain that this is simply integrating the density $|\omega|$. Can anyone confirm this? The only fact specific to $K$-analytic manifolds that I used is this ability to write a Borel set as a disjoint union of Borel sets in coordinate patches. That being said:
I believe this requirement that Borel sets are disjoint unions of Borel sets in coordinate patches is actually a red herring. Is it simply true that one can do this? (I don't see how to do this if the patches are not disjoint.) Otherwise, is there a work-around?