  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.
