Okay, again, I was being imprecise. All continuous uniform distributions have support with nonzero finite measure. The discrete uniform distribution is often thought of as a different distribution than the continuous uniform distribution.

This just goes to show how much math relies on people knowing which definitions you happen to be using at the moment you say something.

The point is that not all sets with nonzero finite measure are bounded, therefore you can have a uniform distribution whose support is equal to such a set.