Suppose $\Phi$ is a stationary point process on $\mathbb{R}^d$ under $P$ with intensity $0<\lambda<\infty$. In particular the first moment measure $E[\Phi(B)] = \lambda |B|$ where $|B|$ denotes volume. Denote $P^0$ the Palm measure of $\Phi$. Does it hold that $E^0[\Phi(B)] < \infty$ for all $B$ bounded? Taking $B:=B(0,r)$, for any $\epsilon > 0$ $$ E^0[\Phi(B)] = \frac{1}{\lambda |B(0,\epsilon)|}E\int_{B(0,\epsilon)} \Phi(B(t,r))\,\Phi(dt) \leq \frac{1}{\lambda |B(0,\epsilon)|}E[\Phi(B(0,\epsilon))\Phi(B(0,r+\epsilon)]. $$

Certainly if $E[\Phi(B(0,r +\epsilon)^2]<\infty$ then the previous expectation would be finite, but I would like to avoid this assumption if possible.

Note that if this is true for an arbitrary $B$, one needs "bounded" and not just $|B|<\infty$. E.g. if $\Phi$ is a stationary copy of $\mathbb{Z}^2$ in $\mathbb{R}^2$, then $x$-axis is a set of measure zero that under $P^0$ contains infinitely many points almost surely.