The problem of equal parts
A topological group is extremely amenable if it has a fixed
point under every continuous action on a compact Hausdorff space. The
group of measure preserving transformations of the standard
probability space is extremely amenable by a result of Giordano and
Pestov applying analytical techniques. This in turn implies that the
class of finite measure algebras posseses the approximate Ramsey
property. Via discretization, Giordano and Pestov's result would
follow from (an exact) Ramsey property for finite measure algebras