This isn't possible for some manifolds. In particular, you can't choose a point uniformly from an infinite plane. To see why, split the plane into an infinite number of equally-sized (say, one by one) squares. Each square should have an equal probability that one of its points is chosen. Call this probability s.

Now let's look at the entire plane. If you pick a point, it will surely be on the plane. That means the probability that you choose a point on the plane is equal to one.

Putting the squares together gives you the whole plane, so you'd expect that summing their overall probabilities s would give you the overall probability of the plane, which is one. But there are an infinite number of squares, and there's no real number s for which the sequence s, s + s, s + s + s, ... converges to one.