Piron's and Bell's geometric Lemmas and Gleason's theorem

We study the idea of implantation of Piron's and Bell's geometrical lemmas for proving some results concerning measures on finite as well as infinite- dimensional Hilbert spaces, including also measures with infinite values. I n addition, we present parabola based proofs of weak Piron's geometrical an d Bell's lammas. These approaches will not used directly Gleason's theorem , which is a highly non-trivial result.