Savage's theorem with atoms

The famous theorem of Savage is based on the richness of the states space, by assuming a \textit{continuum} nature for this set. In order to fill the gap, this article considers Savage's theorem with discrete state space. The article points out the importance the existence of pair event in the existence of utility function and the subjective probability. Under the discrete states space, this can be ensured by the intuitive \textit{atom swarming} condition. Applications for the establishment of an inter-temporal evaluation \emph{\`a la } Koopman \cite{K60}, \cite{K72}, and for the configuration under \textit{unlikely atoms} of Mackenzie \cite{Mackenzie2018} are provided.