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Ok so first to have a uniform distribution we have to have a bounded set. Maybe you can do something clever with limits but lets not overcomplicate things. Lets say we have 0 <= v < 10. Define E = v^2. Then 0 <= e < 100

Uniformity of v would mean that p(0 <= v < 1) = 1 / 10

Uniformity of E would mean that p(0 <= E < 1) = 1 / 100

But by construction p(0 <= v < 1) = p(0 <= E < 1). So it's not possible for both to be uniform.

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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.

(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.

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