= Solution
Write $S(t)=\mathbb P(T>t)$ for the <survival function>; this is the function denoted $F_T$ in the survival notation. For a proper continuous event-time distribution with <hazard function> $h$, its <cumulative hazard function> is $H(t)=\int_0^th(s)ds=-\log S(t)$. The <probability integral transform> makes $S(T)$ uniform on $(0,1)$, including when $S$ has flat intervals: those intervals have zero event <probability>. Consequently
$$
\mathbb P(H(T)>u)=\mathbb P(S(T)<e^{-u})=e^{-u},\qquad u\geq0.
$$
Thus the <cumulative hazard probability transformation> gives $\boxed{H(T)\sim\operatorname{Exponential}(1)}$. No strict monotonicity of $H$ is needed on intervals that carry no event mass. The usual proper continuous survival law is essential; an atom of subjects who never fail would instead require separate treatment of the mass at infinity.
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