tinuous is that mass points need not be excluded; distributions representing fractal measures such as the Cantor distribution are also allowed. This means that mixtures of discrete and continuous distributions e.g. such as those ex- amined by An and Hu (2012) for measurement error in survey responses, some of which may be error-contaminated, but some may be truthful lead- ing to a mixture with a mass point distribution are included. Moreover, in S ∗ the case of mass points in the distribution of the mismeasured variable is also easily handled; in the literature such mass points are documented for income or work hours distributions in the presence of rigidities such as unemployment compensation rules (e.g. Green and Riddell, 1997). Fractal distributions may arise in some situations, e.g. Karlin’s (1958) example of the equilibrium price distribution in an oligopolistic game. For regression functions the assumption g ∈ S ∗ implies that growth at infinity is allowed but is somewhat restricted. In particular for any ordinary point-wise defined function b ∈ S ∗ the condition needs to be satisfied for some non-negative valued m$_{1}$, ..., m$_{d}$. If a locally integrable function g is such that its growth at infinity is majorized by a polynomial, then b ≡ g satisfies this condition. While restrictive this still widens the applicability of many currently available approaches. For example in Berkson regression the common assumption is that the regression function 11 ∫ ... ∫ Π d i =1 ( (1 + t 2 i ) − $^{1}$) m i | b ( t ) | dt$_{1}$...dt$_{d}$ < ∞ , (1)