Compound random measures (CoRM) represent a unifying framework for vectors of dependent completely random measures. They are used as priors in a Bayesian nonparametric setting. The definition of Compound Random Measures implies the specification of a score distribution and a directing L\`evy Measure. In this note we provide conditions on the directing L\`evy measure and the score distribution in order to guarantee the existence of the marginal L\`evy intensities. On the other hand, we also provide conditions for the existence of the multivariate L\`evy intensity of a CoRM. The last section is devoted to show some interesting properties of Compound Random Measures when exponential scores and regularly varying L\`evy intensities are considered.
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