This question arose from a real analysis question, where i am more generally looking for a finitely additive measure on $2^{\mathbb{R}}$ for a fixed point $a$ that satisfies the property:
$\forall A \in 2^{\mathbb{R}}: a\notin \overline{A-\{a\}} \implies P(A)=0$
Since an ultrafilter is a finitely additive 0-1-measure and restricting the measure to 0 and 1 would give nice properties to the core question, an ultrafilter satisfying above property would be amazing. However since the property ensures the ultrafilter has to be free (finite sets don't have accumulation points), an explicit construction in ZF is impossible. Is the existence of such a filter assuming AC possible?