Let $\xi$ be a Poisson Random Measure of intensity $\mu$ (informally $\mathbb E\xi = \mu$).


(For $f \ge 0$, say) when does $\xi f = \infty?$ Kallenberg (Foundations of Modern Probabilility) claims the following:


* $\xi f$ exists iff $\mu(|f| \wedge 1) < \infty$.


It's certainly a basic result on Poisson Random Measure that $\mu(|f| \wedge 1) < \infty$ implies $\xi f < \infty$ a.s.. However, he doesn't actually seem to prove the converse, and I'm not sure what he means by the converse either. It could be either of


* $\mathbb P(\xi f < \infty) = 1$ implies $\mu(|f| \wedge 1) < \infty$
* $\mathbb P(\xi f < \infty) > 0$ implies $\mu(|f| \wedge 1) < \infty$


although perhaps these two are equivalent by Kolmogorov's 0-1 law.


Can anyone shed any light on how to prove the converse, or point me to a better reference for Poisson Random Measure?



Non-existence of integral with respect to Poisson Random Measure
As I mentioned in my comment, you can prove the statement and its converse by looking at the moment generating function. Supposing that f ≥ 0 and λ > 0 is a real number, the following is true for a Poisson point measure ξ with Eξ = μ,


$$\mathbb{E}\left[e^{-\lambda\xi f}\right]=\exp\left(-\mu\left(1-e^{-\lambda f}\right)\right).$$


You can calculate the probability that ξf is finite from this using monotone convergence,


$$\mathbb{P}\left(\xi f \lt \infty\right)=\lim\_{\lambda\downarrow 0}\exp\left(-\mu\left(1-e^{-\lambda f}\right)\right).$$


If μ(f∧1) <∞ then (1-e-λf) ≤ f∧1 for all λ ≤ 1, so dominated convergence gives μ(1-e-λf) → 0 as λ →0. So, E[e-λξf] → 1. This gives ξf < ∞ almost surely.
Now, for the converse statement: If μ(f∧1) = ∞ then, using (1-e-λf) ≥ ½ λ(f∧1) for λ ≤ 1 shows that μ(1-e-λf) = ∞. So, E[e-λξf]=0, giving ξf = ∞ almost surely.


And, yes, the Kolmogorov-zero one law does indeed imply that ξf < ∞ with probability 0 or 1. Splitting the space up into a countable sequence of measurable sets Sn on which both μ and f are bounded, then we want to know if the sum ∑nξ(1Snf) of independent random variables is finite, which is a tail event.


Looking at my copy of Kallenberg I see that the statement you give, and I have just proven above, is Lemma 12.13. Precisely quoting his proof: 



> 
> If ξ|f| < ∞ a.s. then μ(|f|∧1) < ∞ by Lemma 12.2. The converse implication was established in the proof of the same lemma.
> 
> 
> 


Looking at his Lemma 12.2, it is the statement of the moment generating function which I just used. Like you say, it doesn't seem like he does establish the converse implication at all. However, it is a relatively simple step using my argument above.


