Regularity properties of measurable functions satisfying a multiplicative type functional equation almost everywhere

摘要

For functional equations originating from probability theory the equation is satisfied for the density functions only almost everywhere by the nature of the method used to obtain it. There are standard theorems proving that all measurable solutions can be extended uniquely to continuous solutions, but in some cases-for example, for multiplicative equations-these can be applied only if we also prove that the solutions are nonzero, too. We shall consider such an example and prove a general theorem as well.