What can be said about the uniqueness of the following integral transformation:
$ (Tf)(u) = \int_0^{\infty} f(t)G(tu)dt$ defined for all $u\geq 0$,
where the kernel $G(z) \in [0,1]$ for all $z\geq0$, and it is monotonically decreasing from 1 to 0 as its argument goes from 0 to $\infty$.
Let $f(t)$ and $h(t)$ are two probability distributions. Suppose they have the same transform, i.e., $ (Tf)(u) = (Th)(u)$ for all $u\geq 0$? Does this imply uniqueness, i.e. $f(t)=g(t)$ for all $t\geq0$?