Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval

Let $a<b$ be real numbers, let $\mathcal{B}$ be the Borel $\sigma$-algebra on $\mathbb{R}$, and let $\lambda$ be Lebesgue measure, whose domain is $\mathcal{B}$ by claim 3 of Existence of Lebesgue Measure on the Real Line. 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 Lebesgue integrals of nonnegative measurable functions and Lebesgue integrals of integrable functions are as in those definitions. Then:

1. (Restricted measure space) $\mathcal{B}_{[a,b]}$ is a $\sigma$-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 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 probability space.

2. (Zero extension) A function $f:[a,b]\to[0,\infty]$ is measurable from $([a,b],\mathcal{B}_{[a,b]})$ to $[0,\infty]$ with its Borel $\sigma$-algebra, in the sense of Lebesgue Integral of a Nonnegative Measurable Function, 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.

3. (Continuous integrands) Regard $[a,b]$ as a subset of the real line with the absolute value metric, and let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$. Then $f$ is Riemann integrable on $[a,b]$, it is a square-integrable random variable on the probability space of claim 1, and its Lebesgue integral agrees with its Riemann integral:

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 $\bigl(\int_{[a,b]}|f|\,d\lambda_{[a,b]}\bigr)^{2}\le(b-a)\int_{[a,b]}f^{2}\,d\lambda_{[a,b]}$.

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 $\lambda_{[a,b]}(\{t\in[a,b]:f(t)>0\})=0$.

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 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 $D$,