Suppose f and g are two Borel function on [0, 1]. The push-forward of the Lebesgue measure on [0,1] by f and by g are the same. Then are there some Borel measurable function from [0,1] to [0,1], such that
f = g(h)?
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Sign up to join this communitySuppose f and g are two Borel function on [0, 1]. The push-forward of the Lebesgue measure on [0,1] by f and by g are the same. Then are there some Borel measurable function from [0,1] to [0,1], such that
f = g(h)?
Assuming push-forward means this ...
Hint... Consider the two maps $f(x) = \{2x\}$ and $g(x) = \{3x\}$, where the brackets are the fractional part. What would $h$ be then?