[SomeStupidPoint]

It's an interesting exercise in measure to figure out what falls apart between a uniform distribution on [0,1] and the lack of one on [0, inf]. (Adding a point at infinity to compactify the set, which makes the two intervals topologically equivalent.)

[klodolph]

To elaborate, the reason topological equivalence doesn't help here is because probability is defined in terms of measure spaces, not topological spaces.