Let $S$ be a countable discrete set, the following two results are quite easy to prove:
Every countably additive probability measure $\mu$ on $S$ commutes (in Fubini's sense) with every finitely additive probability measure $\nu$ on $S$, i.e. $$ \int\int f(x,y)d\mu(x)d\nu(y)=\int\int f(x,y)d\nu(y)d\mu(x) $$ for all bounded $f$.
The set of countably additive probability measures on $S$ is dense in the weak* topology in the set of all finitely additive probability measures on $S$.
My question is:
Question: Do those results hold also for uncountable sets?
More precisely, I am interested in the case when $S$ is a locally compact group and the measures are identified with positive functional on $L^\infty(S)$, and the latter is defined with respect to the Haar measure.
Update: Question 2., in its natural setting, as positive answer, as observed in the comments by Bill Johnson. Question 1. is still unanswered.
Thanks in advance,