It is known that for a compact metric space $X$ without isolated points the set of nonatomic Borel probability measures on $X$ is dense in the set of all Borel probability measures on $X$ (endowed with the Prokhorov metric). In particular if $X$ is a product space $X=X\_1\times\cdots\times X\_n$ (each $X\_i$ a compact metric space), and given a measure $\mu$ on $(X,\mathcal B(X))$ ($\mathcal B(X)$ the Borel subsets of $X$), there is a nonatomic $\nu$ measure on $(X,\mathcal B(X))$ arbitrarily close to $\mu$. In general, $\nu$ need not have nonatomic marginal probability measures (here the marginal for the $i$-th factor is $\nu(X\_1\times\cdots\times X\_{i-1}\times\cdot\times X\_{i+1}\times\cdots\times X\_n)$). Is it known whether a there exists a $\nu$ with nonatomic marginals arbitrarily close to $\mu$?



Nonatomic probability measures
I think that Dave's argument (as well as the reference to the Hilbert cube) make this question more complicated than it actually is. 


Let's take for a starting point the claim already formulated by the topicstarter: for a compact metric space $X$ without isolated points the set of non-atomic Borel probability measures on X is weak$^\*$ dense in the set of all Borel probability measures on $X$. In particular, any delta-measure on $X$ can be approximated by non-atomic measures. It implies that in the case of a product space $X=X\_1\times\dots\times X\_n$ any delta-measure on $X$ can be approximated by products of non-atomic measures (so that, in particular, all their marginals are non-atomic). Now, in turn, an arbitrary measure on $X$ can be approximated by finite convex combinations of delta-measures on $X$. 


