I'm really new to this area of random measures, and I'm a bit confused on how to get started on this problem.
Let $\mu$ be a measure on $\mathbb{R}$ with $\mu(\left\{0\right\}) = 0$ and $\int_\mathbb{R} |x| \wedge 1 \mu(dx) < \infty$.
Define $\nu = \lambda \times \mu$.
Let $M$ be a Poisson Random Measure with mean $\nu$. Let $Z_t = \int_{[0,t] \times \mathbb{R}} M(ds,dx)$.
Prove that $Z_t < \infty$ a.s., i.e. $\mathbb{P}(Z_t<\infty)=1$.
I'm not really sure how to approach this problem. I managed to show that $\nu$ must be $\sigma$-finite. I also know the following:
$$P(Z_t < \infty) = lim_{\alpha \rightarrow 0} E[e^{\alpha Z_t}]\ \text{and}\ Z_t = lim_{m \rightarrow \infty} \int_{[0,t] \times (\frac{1}{m}, \infty]} M(ds,dx).$$
Normally the way I usually show something is finite a.s. is to show it has finite expectation a.s., but I don't see a way to make that work in this case.