Let $a<b$ be \reftext{def:real-numbers-c54-2026c}{real numbers}, let $\mathcal{B}$ be the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra} on $\mathbb{R}$, and let $\lambda$ be \reftext{thm:lebesgue-measure-real-line-2026a}{Lebesgue measure}, whose domain is $\mathcal{B}$ by claim 3 of \ref{thm:lebesgue-measure-real-line-2026a}. Define

and for $f:[a,b]\to\mathbb{R}$ let $\tilde f:\mathbb{R}\to\mathbb{R}$ denote the zero extension, $\tilde f=f$ on $[a,b]$ and $\tilde f=0$ elsewhere, and likewise for $[0,\infty]$-valued $f$. All \reftext{def:lebesgue-integral-nonnegative-2026a}{Lebesgue integrals of nonnegative measurable functions} and \reftext{def:lebesgue-integral-integrable-2026a}{Lebesgue integrals of integrable functions} are as in those definitions. Then:

\textbf{1. (Restricted measure space)} $\mathcal{B}_{[a,b]}$ is a \reftext{def:sigma-algebra-measurable-space-2026a}{$\sigma$-algebra} on $[a,b]$, every member of $\mathcal{B}_{[a,b]}$ belongs to $\mathcal{B}$, and $([a,b],\mathcal{B}_{[a,b]},\lambda_{[a,b]})$ is a \reftext{def:measure-measure-space-2026a}{measure space} with $\lambda_{[a,b]}([a,b])=b-a$. Consequently $([a,b],\mathcal{B}_{[a,b]},(b-a)^{-1}\lambda_{[a,b]})$ is a \reftext{def:probability-space-random-variable-2026a}{probability space}.

\textbf{2. (Zero extension)} A function $f:[a,b]\to[0,\infty]$ is \reftext{def:measurable-function-2026a}{measurable} from $([a,b],\mathcal{B}_{[a,b]})$ to $[0,\infty]$ with its Borel $\sigma$-algebra, in the sense of \ref{def:lebesgue-integral-nonnegative-2026a}, if and only if $\tilde f$ is so measurable from $(\mathbb{R},\mathcal{B})$; and in that case

The same holds for real-valued $f$ with measurability read via $\mathcal{B}$ on $\mathbb{R}$, and integrability of $f$ equivalent to integrability of $\tilde f$, with equal integrals.

\textbf{3. (Continuous integrands)} Every \reftext{def:continuity-closed-interval-c54-2026b}{continuous} $f:[a,b]\to\mathbb{R}$ is a \reftext{def:square-integrable-mean-square-2026a}{square-integrable} \reftext{def:probability-space-random-variable-2026a}{random variable} on the probability space of claim 1, and its Lebesgue integral agrees with its \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral}:

\textbf{4. (Cauchy-Schwarz inequality)} If $f,g:[a,b]\to\mathbb{R}$ are $\mathcal{B}_{[a,b]}$-measurable and $\int_{[a,b]}f^{2}\,d\lambda_{[a,b]}$ and $\int_{[a,b]}g^{2}\,d\lambda_{[a,b]}$ are finite, then $fg$ is integrable with respect to $\lambda_{[a,b]}$ and

In particular, taking $g=1$: $|f|$ is integrable and $\bigl(\int_{[a,b]}|f|\,d\lambda_{[a,b]}\bigr)^{2}\le(b-a)\int_{[a,b]}f^{2}\,d\lambda_{[a,b]}$.

\textbf{5. (Null integrands)} If $f:[a,b]\to[0,\infty)$ is $\mathcal{B}_{[a,b]}$-measurable and $\int_{[a,b]}f\,d\lambda_{[a,b]}=0$, then $\lambda_{[a,b]}(\{t\in[a,b]:f(t)>0\})=0$.

\textbf{6. (Limits on co-null sets)} Let $D\in\mathcal{B}_{[a,b]}$ satisfy $\lambda_{[a,b]}([a,b]\setminus D)=0$. If $f_n:[a,b]\to[0,\infty)$ are $\mathcal{B}_{[a,b]}$-measurable, $f:[a,b]\to[0,\infty)$ satisfies $f(t)=0$ for $t\notin D$, and for every $t\in D$ the real sequence $(f_n(t))_{n}$ has \reftext{def:limit-sequence-real-c54-2026a}{limit} $f(t)$, then $f$ is $\mathcal{B}_{[a,b]}$-measurable. Moreover, for every $\mathcal{B}_{[a,b]}$-measurable $g:[a,b]\to[0,\infty]$, writing $\mathbf{1}_D$ for the indicator of $D$,