Uniform distribution dependent on the parameters?

Question

Say for example I had the unit circle. Then say that I "distribute a unit of probability mass on this circle." I know this means that given any two finite intervals of the same size within the domain of the circle, there lies equal probability in these intervals. However, I'm not sure how this translates into different coordinate systems. For example, I could use a polar coordinate system and distribute it over theta from (0, 2pi] at a fixed r=1. Alternatively, I could use the x-y plane and distribute it that way which would give a different result, right?

Answer 1

 

 The uniform distribution by definition assigns the same probability to any element in the domain.

 

 So if you took any positive area A from the x-y plane and two intervals within that area having

the same length, the probability is going to be the same.

 

Proof:

 

      Given Area A and two sub areas D1 and D2 from A such that |D1|=|D2|,

 

      Prob( D1) = |D1|/A = |D2|/A = Prob(D2)

 

Anything else is not the uniform distribution