Gronwall's Lemma for Bounded Measurable Functions

Let $\mathbb{R}$ be the real numbers, and let $T$, $a$, and $c$ be real numbers with $T>0$ and $c\ge0$. Write $[0,T]$ for the closed interval determined by $0$ and $T$, and likewise $[0,t]$ for the closed interval determined by $0$ and $t$ when $t\in[0,T]$. 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

$$u(t)\le a+c\int_{[0,t]}u(s)\,ds\qquad\text{for every }t\in[0,T],$$

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

Then, with $\exp$ the exponential function, $u(t)\le a\exp(ct)$ for every $t\in[0,T]$.