Take the set of all probability measures over $[0,1]$. Call this $\Delta [0,1]$. Consider an open ball of distributions in $\Delta [0,1]$, in the weak* topology. My question is, must this open ball contain a density with full support? In other words, for any distribution $\mu \in \Delta [0,1]$, does there exist another close to it which is absolutely continuous and has full support?
My approach: I think it definitely needs to have full support. If not, take its $\epsilon$-convex combination with the uniform distribution. For small enough $\epsilon$ this must lie inside the open ball. Now suppose the CDF is discontinuous (i.e. has an atom). It is an increasing function, so can only have jump discontinuities.
Now I think there is a result that says there is a continuous function close to every discontinuous function and a differentiable function close to every non-differentiable one. I should've known this result clearly already but I don't. Any reference for this one would be appreciated as well. (I'm a social sciences student, hence the gaps in basic knowledge simultaneously with the need to use results like these. :( )
Assuming the above result is true, this makes me think we indeed must have a density. But that result uses a different topology, and I'm confused if it is applicable here.
Any help is most appreciated.