In David Applebaum's book Levy Processes and Stochastic Calculus, it defines stochastic integral with respect to martingale-valued measure(roughly speaking, a random measure on $\mathbb{R}^+\times E$ that makes $M([0,t],A)$ is a martingale for each A in $\mathbb{B}(E)$. Here, $E$ is a Borel set in $\mathbb{R}^d$).
The most important martingale-valued measure is the compensated poisson random measure. However, the compensated measure can be well defined only when $0\notin\bar{E}$. Otherwise it causes $\infty-\infty$ which is meaningless. See this question for more details.
And right now I am quite confused that how can we define stochastic integral with respect to compensated poisson random measure. Should we use approximation method in some sense?