On page 2 of "Spatial and Spatio-Temporal Log-Gaussian Cox Processes: Extending the Geostatistical Paradigm" by Diggle et al. (2013), accessible at https://arxiv.org/pdf/1312.6536.pdf , 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 Diggle et al's 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.