Integration by Parts for Indefinite Lebesgue Integrals on a Compact Interval

Let $T>0$ be a real number, and let $f,g:[0,T]\to\mathbb{R}$ be measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and Lebesgue integrable over $[0,T]$. Let $u_0$ and $v_0$ be real numbers and define

Then:

(i) The functions $t\mapsto u_t$ and $t\mapsto v_t$ are continuous on $[0,T]$, the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric, and there is a real number $C\ge0$ with $|u_t|\le C$ and $|v_t|\le C$ for all $t\in[0,T]$.

(ii) The functions $t\mapsto f(t)\,v_t$ and $t\mapsto u_t\,g(t)$ are measurable and Lebesgue integrable over $[0,T]$, and