Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval

In the settings of Measurable Functions and the Lebesgue Integral: Notation and The Real Line: Standing Notation and Background for Calculus on Intervals, assume the axiom of dependent choice. Let $a,b\in\mathbb{R}$ with $a<b$, let $[a,b]$ be the closed interval, and let $\lambda$ be Lebesgue measure on $\mathbb{R}$. Write $([a,b],\mathcal{B}_{[a,b]},\lambda_{[a,b]})$ for the restriction of $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ to $[a,b]$, as in Measurable Functions and the Lebesgue Integral: Notation (ix); thus $\mathcal{B}_{[a,b]}=\{B\in\mathcal{B}(\mathbb{R}):B\subseteq[a,b]\}$ and $\lambda_{[a,b]}(B)=\lambda(B)$ for $B\in\mathcal{B}_{[a,b]}$ (Integrals of Functions over a Measurable Subset (i)). Let $\mathcal{B}([a,b])$ be the Borel $\sigma$-algebra of the metric space $([a,b],d_{[a,b]})$ of The Real Line: Standing Notation and Background for Calculus on Intervals (iv). For functions $[a,b]\to\mathbb{R}$, measurable refers to $\mathcal{B}_{[a,b]}$, and $\mathcal{L}^{1}(\lambda_{[a,b]})$ and $\int f\,d\lambda_{[a,b]}$ are as in Measurable Functions and the Lebesgue Integral: Notation (vi) for this measure space. A function $f:[a,b]\to\mathbb{R}$ is bounded if there is $B\in\mathbb{R}$ with $|f(t)|\le B$ for every $t\in[a,b]$.

(i) The measure space. The measure $\lambda_{[a,b]}$ is finite, with

$$\lambda_{[a,b]}([a,b])=b-a,$$

and $\mathcal{B}_{[a,b]}=\mathcal{B}([a,b])$.

(ii) Riemann integral and Lebesgue integral. Let $f:[a,b]\to\mathbb{R}$ be bounded, Riemann integrable on $[a,b]$ and measurable. Then $f\in\mathcal{L}^{1}(\lambda_{[a,b]})$ and

$$\int f\,d\lambda_{[a,b]}=\int_{a}^{b}f,$$

the right-hand side being the Riemann integral of $f$ over $[a,b]$.

(iii) Continuous functions. Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$. Then $f$ is measurable, bounded and Riemann integrable on $[a,b]$, $f\in\mathcal{L}^{1}(\lambda_{[a,b]})$, and

$$\int f\,d\lambda_{[a,b]}=\int_{a}^{b}f .$$