The Entropy of Random Finite Sets

We define the Boltzmann-Gibbs entropy of random finite set on a general space X as the integration of logarithm of density function over the space of finite sets of X , where the measure for this integration is the dominating measure over this space. We show that with a unit adjustment term, the same value for entropy can be obtained using the calculus of set integrals which applies integration of densities that have unit. Extending this concept of entropy into conditional entropy and mutual information, we use the fundamental result on the relation of the mutual information and the variance of filtering error to derive a counterpart result for the signals and systems with finite set nature in a special case. While the expression for the BoltzmannShannon entropy of discrete Poisson distribution is by an infinite series, we show that for a uniform Poisson random finite set which has Poisson cardinality distribution with mean of λ, the entropy is equal to λ nat.