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Inference on linear functionals of the latent distribution in standard measurement error models is considered. The question about asymptotically efficient estimation by maximum likelihood in a convolution model with Laplace error distribution is settled in the affirmative: maximum likelihood estimators of certain linear functionals are \sqrt{n}-consistent, asymptotically normal and efficient, hence equivalent to simple naive estimators that are empirical means of a given transformation of the observations. Asymptotic normality of a Studentized version of naıve estimators allows to construct asymptotic confidence intervals for linear functionals. Regarding maximum likelihood estimation of the mixing distribution as a data-driven choice of the a priori distribution on the mixing parameter in an empirical Bayes approach to the problem of estimating the single means, a sequence of estimators can be constructed that is asymptotically optimal in a decision-theoretic sense.