In the paper is given a general definition of the entropy of a probability density. Then convex sets of probability densities are constructed in which exist densities of maximum entropy. Furthermore it is demonstrated that from the convergence of the entropy of a sequence of probability densities being contained in the convex set R against the entropy of distribution of the maximum entropy follows the convergence of the appropiate probability measures against each other with reference to the total veriation. Hence CSISZAR's results will be taken as an applications and a sufficient condition is given for the convergence against the normal, exponential and geometrical distribution.