We find infinite densities, which describe strong anomalous diffusion where <|x(t)|^q>~t^{q ν(q)} with a nonlinear spectrum ν(q)≠const. The latter is widespread and found in nonlinear deterministic dynamics and experiments of active transport in live cells. Using a stochastic approach we show how this phenomena is related to the infinite densities, i.e., in the long time limit the system is described by a non-normalized state. Our work shows that infinite measures play an important role in the statistical description of open systems exhibiting multi-fractal anomalous diffusion as they are complementary to the central limit theorem.