In this note we study the relation between belief functions of Dempster-Shafer theory and inner measures induced by probability functions. In
[3,4] Joe Halpern and Ron Fagin claim that these classes of functions are
essentially the same, or, more precisely, that they are exactly the same in
case the functions are defined on formulas rather than sets. We show that, when the functions are defined on sets only a proper subclass off the belief
functions over aframe S corresponds to the class of inner measures induced
by a probability measurer on some algebra, on S. However, belief functions,
over S do correspond to inner measures induced by probability measures
defined on algebras on refinements of S. fact that in general refinemeats
of S are needed to obtain all belief functions over S is shown to be
obscured by the particular way formulas are assigned probabilities or
weights in [3]
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