Does the distribution of $|f|$ coincides with that of $|g|$?

Question

Let $f,g$ be two (Lebesgue) measurable function on $[0,1]$. Assume that $$\int \frac{\alpha |f(m)|}{1+\alpha |f(m)|}dm =\int \frac{\alpha |g(m)|}{1+\alpha|g(m)|}dm $$ for any $\alpha \ge 0$. Does the distribution of $|f|$ coincide with that of $|g|$?