Full measure properties for Zariski open subsets in $p$-adic situation

Let $F$ be a $p$-adic field and let $X$ be a smooth integral variety over $F$ (I am chiefly interested in the case when $X$ is a connected reductive group over $F$). Let $U$ be a non-empty open subset of $X$ with complement $Z$.

We can endow $X(F)$ with the Serre–Oesterlé measure (e.g., as in [1, Section 2.2] or [2, Section 7.4])—this is just the standard measure coming from a top form of $X$).

My question is then whether one knows a simple proof/reference for the following:

Claim: The subset $Z(F)$ of $X(F)$ has measure zero.

I think this is proven in [1, Lemma 2.14]—but this is concerned with a more specific context which makes it non-ideal as a reference.

Any help is appreciated!

[1] Magni – $p$-adic integration and birational Calabi–Yau varieties.

[2] Igusa, J.I., 2007. An introduction to the theory of local zeta functions (Vol. 14). American Mathematical Soc.