Ulam's theorem is stated on page 25, Measure and Category(2ed), John Oxtoby, as:
A finite measure $\mu$ defined for all subsets of $X$ of power less than the least weakly inaccessible cardinal vanishes identically if it equals zero for every one-element subset.
I saw Sierpiński's theorem in an answer of MO.
If $μ:{\mathcal {P(Ω)}}→[0,1]$ is a probability measure and $|Ω|$ is smaller than the first weakly-inaccessible cardinal, then there must be a countable $A⊆Ω$ such that $μ(A)=1$.
I don't have a reference on the second theorem, though I saw it has been mentioned several times on this site. It seems to me they're telling the same thing.
Sierpiński's theorem must assume every singleton is assigned with a measure of zero execpt for countable many of singletons in $\Omega$, or it can't be a finite measure. Let $A$ be a countable subset of $Ω$ which is the union of all singletons with non-zero measure, Ulam's theorem imply that $Ω \setminus A$ must "vanishes identically". Thus the probability measure must concentrate on $A$. The reverse reasoning is more or less the same.
Probably I misunderstood something obvious, otherwise they can't be named after by two Polish mathematicians who seem to know each other well.