  In this paper, we consider certain $\sigma$-finite measures which can be
interpreted as the output of a linear filter. We assume that these measures
have regularly varying tails and study whether the input to the linear filter
must have regularly varying tails as well. 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
i.i.d. random variables, to products of independent random variables, and to
stochastic integrals with respect to L\'evy motions.
