We tackle the inverse problem of reconstructing an unknown finite measure $μ$ from a noisy observation of a generalized moment of $μ$ defined as the integral of a continuous and bounded operator $Φ$ with respect to $μ$. When only a quadratic approximation $Φ_m$ 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.