  We tackle the inverse problem of reconstructing an unknown finite measure
$\mu$ from a noisy observation of a generalized moment of $\mu$ defined as the
integral of a continuous and bounded operator $\Phi$ with respect to $\mu$.
When only a quadratic approximation $\Phi_m$ of the operator is known, we
introduce the $L^2$ approximate maximum entropy solution as a minimizer of a
convex functional subject to a sequence of convex constraints. Under several
assumptions on the convex functional, the convergence of the approximate
solution is established and rates of convergence are provided.
