> In particular, you can't choose a point uniformly from an infinite plane. While this is true, your reasoning is incomplete, you only arrive at the conclusion

> In particular, you can't choose a point uniformly from an infinite plane. While this is true, your reasoning is incomplete, you only arrive at the conclusion that s couldn't be positive. Why couldn't be s equal to 0? Note that a 0 probability event is not an impossible event and could still be in the event space. The problem is that there are countably infinite squares on the infinite plane so the measure of the infinite plane would add up to 0 instead of 1 by the property of the measure of countably infinite union of measurable sets.