[omegalulw]

It's not necessary to have p(0 <= v < 1) = p(0 <= E < 1). Only that P(f(X)) is uniform.

But this does bring up a good point. H(X||Y) = H(f(X)||f(Y)) for any bijective f if the distributions are discrete. When they are continuous this is not true, even with a bijective f. For example f = x^2 doesn't work even though it yields a binary distribution. Interestingly however, affine transformations work.

[im3w1l]

(For non-negative v) v^2 is less than one if and only if v is less than one. That's why the probabilities have to agree in our specific case.