Relation between $\lim_{a \to 0}\int_a^T u(t)$ and the Lebesgue integral $\int_0^T u(t)$

Question

Let $u\colon (0,T] \to \mathbb{R}$ be function with $u \geq 0$ everywhere and $u$ is continuous on $[a,T]$ for every $a > 0$.

Suppose that the limit $$\lim_{a \to 0}\int_a^T u(t) \;dt =U(T)-U(0)$$ exists for function $U$.

I want to know if this is enough to conclude that $\int_0^Tu$, the Lebesgue integral, exists and equals $U(T)-U(0)$. If not, is there some theorem that tells me what I need?