On page 2 of "Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm" by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at arXiv, they claim the following on the bottom right of the page:

"...in the stationary case the intensity of the Cox process is equal to the expectation of Λ(x)"

My assumption is that by "intensity", they are referring to some extension of $$ \lambda (t)=\lim _{h\downarrow 0}{\frac {1}{h}}\mathbb {E} [N(t+h)-N(t)|{\mathcal {F}}_{t}],$$ as taken from wikipedia at https://en.wikipedia.org/wiki/Intensity_of_counting_processes.

I mentioned "extension" because in the Diggle–Moraga–Rowlingson–Taylor paper, they define the Cox process on $\mathbb {R}^2$, whereas the intensity definition from Wikipedia is to do with processes defined on $\mathbb{R}$.

My issue is that I'm unable to find a proof of their claim. If someone more experienced in this field could explain to me why the authors' claim is true / guide me to a relevant source, I would really appreciate it. Thank you.