Soler's theorem and characterization of inner product spaces

Using Soler's result, we show that the existence of at least one finitely additive probability measure on the system of all orthogonally closed subspaces of S which is concentrated on a one-dimensional subspace of E can imply that E is a real, complex, or quaternionic Hilbert space. In addition, using the concept of test spaces of Foulis and Randall and introducing various systems of subspaces of E, we give some characterizations of inner product spaces which imply that E is a real, complex, or quaternionic Hilbert space.