On zero-measured subsets of Thompson's group F

A (discrete) group is called amenable whenever there exists a finitely additive right invariant probablity measure on it. For Thompson's group $F$ the problem whether it is amenable is a long-standing open question. We consider presentation of $F$ in terms of non-spherical semigroup diagrams. There is a natural partition of $F$ into 7 parts in terms of these diagrams. We show that for any measure with the above properties on $F$, all but one of these sets have zero measure. This helps to clarify the structure of Folner sets in $F$ provided the group is amenable.