Let $a<b$ be \reftext{def:real-numbers-c54-2026c}{real numbers} and let $h:[a,b]\to\mathbb{R}$ be \reftext{def:continuity-closed-interval-c54-2026b}{continuous on the closed interval} $[a,b]$. Define the zero extension $\tilde{h}:\mathbb{R}\to\mathbb{R}$ by $\tilde{h}(x)=h(x)$ for $x\in[a,b]$ and $\tilde{h}(x)=0$ otherwise. Then the following hold.

\textbf{Claim 1.} $h$ is \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integrable} on $[a,b]$, by \ref{lem:continuous-implies-riemann-integrable-c54-2026b}.

\textbf{Claim 2.} $\tilde{h}$ is \reftext{def:measurable-function-2026a}{measurable} with respect to the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra} and \reftext{def:lebesgue-integral-integrable-2026a}{integrable} with respect to \reftext{thm:lebesgue-measure-real-line-2026a}{Lebesgue measure} $\lambda$.

\textbf{Claim 3.} The two integrals agree:

where the right-hand side is the Riemann integral of claim 1.