Integration by Parts for Indefinite Lebesgue Integrals on a Compact Interval

Let $T>0$ be a \reftext{def:real-numbers-c54-2026c}{real number}, and let $f,g:[0,T]\to\mathbb{R}$ be \reftext{def:measurable-function-2026a}{measurable} with respect to the \reftext{lem:interval-lebesgue-toolkit-2026a}{trace Borel $\sigma$-algebra} on $[0,T]$ and \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integrable} over $[0,T]$. Let $u_0$ and $v_0$ be real numbers and define

Then:

\textbf{(i)} The functions $t\mapsto u_t$ and $t\mapsto v_t$ are \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on $[0,T]$, and there is a real number $C\ge0$ with $|u_t|\le C$ and $|v_t|\le C$ for all $t\in[0,T]$.

\textbf{(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