On non-atomic weighted majority games

We present some characterizations for the class of non-atomic weighted majority games which are defined on a measurable space (I,C). The characterizations are done within the class of all monotonic simple games which are upper semicontinuous on C and continuous at I with respect to the N A-topology on C. We also use the results on simple games to obtain a characterization for the games of the form f{hook} {ring operator} μ where μ is a non-atomic probability measure and f{hook} is a nondecreasing upper semicontinuous function on [0,1].