A finitely additive measure on projections in a \(W^*\)-algebra \({\mathcal A}\) is a non-negative real valued function \(\mu\) defind on the set \({\mathcal P}\) of all projections in \({\mathcal A}\), that satisfies \(\mu (E+F)=\mu (E)+\mu (F)\) when E,F\(\in {\mathcal P}\) and \(EF=0\). In the paper the following result of ''Gleason type'' is proved. If \(\mu\) is a finitely additive measure on projections in a finite \(W^*\)-algebra \({\mathcal A}\) with no type \(I_ 2\) summand, then there exists a positive linear functional on \({\mathcal A}\) whose restriction to the set \({\mathcal P}\) is \(\mu\).