Molarity can simply be defined as
$$\text{c} \triangleq \frac{n}{V}$$
where $c$ is the molarity, $n$ is the number of moles of a substance of interest (theoretically a counting measure because a given volume contains a finite number of atoms, but concentrations are rarely estimated by 'counting' per se) divided by the volume of the system under consideration (possibly a Lebesgue measure for our purposes).
Holding the volume to be a non-zero constant that we might define as $\alpha \triangleq\frac{1}{V}$ gives us that $c = \alpha n$. Amir Dembo's Stanford University notes on probability theory tell me that a linear combination of measures whose coefficients $\alpha_i \geq 0 \forall i \cdots n$ is also a measure. The issue is that $V$ is not necessarily held to be a constant.
So, after accounting for $V$ being a variable, is the molarity a measure?
I guess you are proposing that molarity as a set function would act on subsets of "physical space".
Moles are a direct count of the number of the chemical species, and would be the closest to a measure. But a molarity which is a measure(in the physics sense) of concentration is not additive; two beakers with 1 litre each of identical fluids, e.g. at $1\ \mathrm c$ when combined (unioned?) is still $1\ \mathrm c$, instead of the $2\ \mathrm c$ demanded by additivity.
Since a measure (in the sense of measure theory) needs to be additive (in fact countably additive), this isn't a measure.