A random (nonnegative Radon) measure $M$ (on $\mathbb R^n$, say) has its law characterized by the Laplace transform $\mathbb E\exp(-\int \varphi(x)\ M(dx))$, $\varphi\in C_c^+(\mathbb R^n)$ (nonnegative continuous functions with compact support), or the characteristic functional (Fourier transform) $\mathbb E\exp(i\int \varphi(x)\ M(dx))$, $\varphi\in C_c(\mathbb R^n)$.
I'm looking for necessary and/or sufficient conditions so that $M$ be purely atomic with probability $1$, and conditions so that $M$ further have finitely many atoms on bounded sets. How are these almost sure properties reflected on the Laplace or Fourier transform (of the law of $M$) ?
If the Laplace transform has the special form $\exp(−∫F(φ(x)) dx)$ , which is when $M(A)$ and $M(B)$ are independent whenever $A\cap B=\emptyset$, then $M$ (nonnegative Radon measure) is atomic (+ a non random constant), and conditions for locally finite support of the atomic part may be expressed in terms of $F$. This is the kind of result I'd like a reference for, as well as extension to more general nonnegative random measures.
