Let $\mathcal{B}([0, 1])$ be the Boolean algebra of measurable subsets of $[0, 1]$ modulo almost everywhere equivalence, i.e., two measurable sets which differ only by a Lebesgue null set are identified. For each $t \in [0, 1]$, let $\mathcal{D}\_t$ be the filter on $\mathcal{B}([0, 1])$ generated by open neighborhoods of $t$. Does these exists a filter $\mathcal{F}$ on $\mathcal{B}([0, 1])$ s.t. for each $t \in [0, 1]$, the filter generated by $\mathcal{D}\_t$ and $\mathcal{F}$ is an ultrafilter?



The existence of a maximal “cross-sectional” filter on the Boolean algebra of measurable subsets of [0, 1] modulo almost everywhere equivalence
No. Suppose $\mathcal{F}$ is such a filter. Clearly each $X \in \mathcal{F}$ has positive measure intersection with every positive-length interval. Then neither $[0,1/2]$ nor $[1/2, 1]$ are in the filter generated by $\mathcal{D}\_{1/2}$ and $\mathcal{F},$ contradiction.


