Law of large numbers for Pengg-expectation

G-frames, which were proposed recently as generalized frames in Hilbert spaces, share many similar properties with frames, but not all the properties of them are similar. Christensen presented that every Riesz frame contains a Riesz basis. In this paper, the authors showed that not all g-Riesz frames contain a g-Riesz basis, but they obtained that every g-Riesz frame contains an exact g-frame. They also gave a necessary and su±cient condition for a g-Riesz frame in a Hilbert space. From this, they might get the characterization of Riesz frames. Lastly the authors considered the stability of a g-Riesz frame for a Hilbert space under perturbations. These properties of g-Riesz frames in Hilbert spaces are not similar to those of Riesz frames.