Can anyone prove this proposition? It is proposition 3.2.2. out of A. Dvurecenskij's book on the applications of Gleason's Theorem. The proof in the book just says that it is obvious and doesn't actually give a proof.
Let m be a completely additive signed measure on L(H). Then a mapping $f_m$ is a frame function on H with weight $m(H)$ is defined by \begin{equation} f_m(x) = m(sp(x)), x \in S(H). \end{equation} Conversely, if f is a frame function on H, where M is a closed subsapace of H and $\{x_i\}$ is an ONB in M, then \begin{equation} m^f(M) := \sum_i f(x_i), M \in L(H), \end{equation} is a completely additive signed measure on $L(H)$, and $f_{m^f} = f$.
In the proposition H is a Hilbert Space, S is the unit sphere on H and L is the quantum Logic.