Let $X_{1}, X_{2}, \ldots$ be random measures and $\lambda_{1}, \lambda_{2}, \ldots \in[0, \infty) .$ Define $X:=\sum_{n=1}^{\infty} \lambda_{n} X_{n} .$ Show that $X$ is a random measure if and only if we have $\mathbf{P}[X(B)<\infty]=1$ for all $B \in \mathcal{B}_{b}(E) .$ Infer that if $X$ is a random variable with values in $(\tilde{\mathcal{M}}(E), \tilde{\mathbb{M}}(E))$ and $\mathbf{E}[X] \in \mathcal{M}(E)$, then $X$ is a random measure.