Gronwall's Lemma for Bounded Measurable Functions

Let $T>0$, $a$, and $c\ge0$ be real numbers, and let $u:[0,T]\to\mathbb{R}$ be bounded and measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and the Borel $\sigma$-algebra on the real line. Assume that

where the integral is the Lebesgue integral over the compact interval $[0,t]$, taken to be $0$ for $t=0$.

Then $u(t)\le a\,e^{ct}$ for every $t\in[0,T]$.