( E ( x ∗ $_{k}$f$_{z}$ ( z ) | z ) , ψ ) = ∫ [ ∫ x ∗ $_{k}$f$_{x}$ ∗ $_{,z}$( x $^{∗}$, z ) dx $^{∗}$] ψ ( z ) dz = ∫ ∫ x ∗ $_{k}$f$_{x}$ ∗ $_{,z}$( x $^{∗}$, z ) ψ ( z ) dx $^{∗}$dz = ∫ ∫ x ∗ $_{k}$ψ ( x ∗ + u ) f$_{x}$ ∗ $_{,u}$( x $^{∗}$, u ) dx $^{∗}$du = ∫ ∫ x ∗ $_{k}$f$_{x}$ ∗ ( x $^{∗}$) f$_{u}$ ( u ) ψ ( x ∗ + u ) dx $^{∗}$du = ( h$_{k}$ ∗ f$_{u}$, ψ ) . The third expression is a double integral which always exists if E ‖ x $^{∗}$‖ < ∞ ; this is a consequence of boundedness of the expectations of z and u. The fourth is a result of change of variables ( x $^{∗}$, z ) into ( x $^{∗}$, u ) , the fifth uses independence of x $^{∗}$and u, and the sixth expression follows from the corre- sponding expression for the convolution of generalized functions (Schwartz, 1967, p.246). The conditions of model 3 are not sufficient to identify the distribution of u$_{x}$ ; this is treated as a nuisance part in model 3. The model in 4 with all the errors and mis-measured variable independent of each other was investigated by Kotlyarski (1967) who worked with the joint characteristic function. In 4 consider in addition to the equations written for model 3 another that uses the independence between x ∗ and u$_{x}$ and involves f$_{u}$$_{x}$. In representation 4a the convolution equations involving the density f$_{u}$$_{x}$ are obtained by applying the derivations that were used here for the model in 3.: z = x ∗ + u ; x = x ∗ + u$_{x}$, 18