In this paper, we develop a refined version of the usual Poisson model for positron emission tomography (PET), in which the data space is finite dimensional, but the unknown emission intensity is represented by a Borel measure on the region of interest. We demonstrate that maximum likelihood (ML) estimators exist in the space of Borel measures and analyse an extension of the finite dimensional EM algorithm for reconstructing the emission intensity. We present evidence that convergence of this functional iteration should be considered in the weak topology and obtain partial convergence results, which contain all the known convergence results to date as special cases. General conditions are obtained under which an ML estimator can be represented by a bounded function. In particular, we show that the regularity of ML estimators depends heavily on properties of the probabilities governing the PET mathematical model. We also show that, in some cases, no ML estimator can be represented by a bounded function. Although this paper is motivated by PET, the results apply to general inverse problems in which the unknown measure, and the kernel representing the blurring operator are all positive.