Note that the OP desired that "each point has an equal probability of being chosen". The infinite plane being infinite, this requirement does not formally make

Note that the OP desired that "each point has an equal probability of being chosen". The infinite plane being infinite, this requirement does not formally make sense, but a natural way to interpret it is to ask for invariance under isometries (or translation-invariance). As the post that you replied to indicates, this is not possible for the Euclidean plane. The measure constructed by the lemma that you cite does not have this property.