I have a set of values. Let's call the set X with values formula ... . Those values in [0, 1] have a non uniform distribution (empirically measured). I would like to re-map those values on [0, 1] such that the distribution of the mapped values is uniform and the values order preserved.

In other words, I would like to find the function f() such that the distribution of f(X) is uniform, and such that iff formula then formula .

What would be a good approach to do that?

There is probably a name for this transformation but I do not know it. I would love to learn it if you know it.