Inverse problems for regular variation of linear filters, a cancellation property for $σ$-finite measures, and identification of stable laws

We study a group of related problems: the extent to which the presence of regular variation in the tail of certain $\sigma$-finite measures at the output of a linear filter determines the corresponding regular variation of a measure at the input to the filter. This turns out to be related to the presence of a particular cancellation property in $\sigma$-finite measures, which, in turn, is related to the uniqueness of the solution of certain functional equations. The techniques we develop are applied to weighted sums of iid random variables, to products of independent random variables, and to stochastic integrals with respect to L\'evy motions.