Sequential densities of rational languages

We introduce the notion of density of a rational language with respect to a sequence of probability measures. We prove that if $(\mu_n)$ is a sequence of Bernoulli measures converging to a positive Bernoulli measure $\overline{\mu}$, the sequential density is the ordinary density with respect to $\overline{\mu}$. We also prove that if $(\mu_n)$ is a sequence of invariant probability measures converging in the strong sense to an invariant probability measure $\overline{\mu}$, then the sequential density of every rational language exists for this sequence.