The Cramér Rao bound provides a minimum achievable variance or covariance for a parameter for a univariate or vector-valued parameter. Point processes often have parameters that are described by functions and the variance and covariance for point processes are themselves functions with spatial variates. Consequently, the usual formulation of the Cramér Rao bound in these contexts is not applicable. The second-order derivative of Kullback’s inequality, which relates the Kullback-Leibler divergence to Cramér rate function, provides a description of the Cramér Rao bound. We follow this approach to develop a form of Cramér Rao bound for point processes and random measures derived from the second-order functional derivative of Kullback’s inequality, which relates the Kullback-Leibler divergence to Cramér’s rate functional for point processes and random measures.