In the measurement error problem the function ε is a characteristic func- tion; if all ε$_{n}$ in the equation (15) are characteristic functions or at least some continuous and uniformly bounded functions then they satisfy the conditions of Lemma 2 with ¯ m = 0. 

The somewhat unusual requirement that the sequence that converges be bounded in S ∗ is a requirement that is associated with the weak topology of the space of generalized functions. In normed spaces convergence implies eventual boundedness in norm of the converging sequence, but this is not the case for generalized functions that do not exclude singularities and polyno- mial growth. The boundedness requirement uniformly limits the degree of singularity and the rate of divergence at infinity of the generalized functions in the bounded set. It is however less restrictive than many assumptions in normed spaces such as boundedness or integrability. 

The Lemma demonstrates that unlike the behavior in function spaces where the deconvolution is usually ill-posed, in the weaker topology of the space S ∗ deconvolution is well-posed when the error distribution is such that the condition (13) is satisfied. Usually well-posedness in deconvolution is examined in terms of ordinary smooth and supersmooth distributions. When the distribution is ordinary smooth the condition (13) holds; it is not satisfied for supersmooth distributions. 

An and Hu (20111) demonstrate that well-posedness holds automatically in a measurement error problem provided there is a mass at zero; this happens if there is a reporting error but with some non-zero probability some values 

