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,

Valerio

`$L^1(S)$`

is weak$^*$ dense in its second dual by Banach-Alaoglu? – Bill Johnson Dec 28 '11 at 14:41