How does one rigorously construct $\mathbb{R}^N$ where $N$ is a $\mathbb{Z}^{++}$-valued random variable on some Borel probability space $(\Omega,\mathcal{B},\mathbb{P})$?
Would someone be so kind to sketch out the main steps in such a construction? I believe that this problem was posed as a thought experiment on one of Terry Tao's blog posts, however I am now unable to find it.