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In this paper, we consider certain $\sigma$-finite measures which can beinterpreted as the output of a linear filter. We assume that these measureshave regularly varying tails and study whether the input to the linear filtermust have regularly varying tails as well. This turns out to be related to thepresence of a particular cancellation property in $\sigma$-finite measures,which in turn, is related to the uniqueness of the solution of certainfunctional equations. The techniques we develop are applied to weighted sums ofi.i.d. random variables, to products of independent random variables, and tostochastic integrals with respect to L\'evy motions.

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