Histogram Equalization
Assuming that in your question $T(r)$ denotes a map $:\mathbb{R}\to[0,1]$ and assuming that it is continuous and strictly increasing, then it is obviously an isomorphism among $r\in[0,1]$ onto its image.
As a consequence you get the following result: let $r$ be any value in $[0,1)$, and let $a=T(r)$, then for any $\rho\in[r,r+dr]$ it is $\alpha = T(\rho) \in [a,a+da]$, having denoted $a+da = T(r+dr)$. Therefore the probability of $r$ and $a$ to belong their infinitesimal intervals (which is actually the definition of PDF) is the same. hence as $dr\to 0$, also $da\to 0$ and you can write
$$
p_r(r) dr = p_a(a) da
$$
which yields the result