  We study the non-parametric estimation of the value ${\theta}(f )$ of a
linear functional evaluated at an unknown density function f with support on
$R_+$ based on an i.i.d. sample with multiplicative measurement errors. The
proposed estimation procedure combines the estimation of the Mellin transform
of the density $f$ and a regularisation of the inverse of the Mellin transform
by a spectral cut-off. In order to bound the mean squared error we distinguish
several scenarios characterised through different decays of the upcoming Mellin
transforms and the smoothnes of the linear functional. In fact, we identify
scenarios, where a non-trivial choice of the upcoming tuning parameter is
necessary and propose a data-driven choice based on a Goldenshluger-Lepski
method. Additionally, we show minimax-optimality over Mellin-Sobolev spaces of
the estimator.
