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. In
addition, we present  parabola based proofs of weak Piron's
geometrical and Bell's lemmas. These approaches will not
used directly Gleason's theorem, which is a highly
non-trivial result.
