On equidistribution of primes in positive characteristic

Question

In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking values in the idèles $J$ of the given number field $k$. Then, under some hypotheses, given $G$ a commutative compact group and $\sigma:J\longrightarrow G$ a continuous homomorphism vanishing on the elements of $k^*$, the set $P$ is equidistributed in $G$ with respect to the map$$\lambda:=\sigma\circ \tau.$$I would expect that a similar result should also hold for $k$ function field in positive characteristic, but I could not find it stated in any of the books I have consulted. I was wondering if somebody is aware of a reference for such a result or if I can be suggested some source I may try to look at before proving it by myself