Proof of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval

Conventions. Write $\mathcal{B}'=\mathcal{B}_{[a,b]}$ and $\lambda'=\lambda_{[a,b]}$.

Proof of (i). The function $\lambda$ is a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ and $[a,b]\in\mathcal{B}(\mathbb{R})$ (Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iv), Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i)), so $\lambda'$ is a measure on $([a,b],\mathcal{B}')$ (Integrals of Functions over a Measurable Subset (i)). As $[a,b]\in\mathcal{B}'$,

$$\lambda'([a,b])=\lambda([a,b])=b-a<\infty$$

(Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i), Small Facts about [0,∞]: Zero, Finiteness and Sums of Zeros (ii)), so $\lambda'$ is finite (Measure, Measure Space, and Probability Measure (iii)). As $d_{[a,b]}$ is the restriction of $d_{\mathbb{R}}$ to $[a,b]\times[a,b]$ and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra of $(\mathbb{R},d_{\mathbb{R}})$, the Borel sets of the subspace $[a,b]$ are $\mathcal{B}([a,b])=\{B\in\mathcal{B}(\mathbb{R}):B\subseteq[a,b]\}=\mathcal{B}'$ (The Real Line: Standing Notation and Background for Calculus on Intervals (iv), Borel Sigma-Algebra of a Metric Space (ii), Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces (vii)).

Proof of (ii). By Darboux Sums Bound the Lebesgue Integral on a Closed Interval (i), $f\in\mathcal{L}^{1}(\lambda')$ and

$$L(f,P)\le\int f\,d\lambda'\le U(f,P)$$

for every partition $P$ of $[a,b]$, with the lower sum $L(f,P)$ and the upper sum $U(f,P)$. So $\int f\,d\lambda'$ is an upper bound of the set $\mathcal{L}(f)$ of lower sums and a lower bound of the set $\mathcal{U}(f)$ of upper sums (The Riemann Integral on a Closed Interval via Lower and Upper Integrals (i), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (ii), Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (i)). The lower integral is the least upper bound of $\mathcal{L}(f)$ and the upper integral the greatest lower bound of $\mathcal{U}(f)$, so

$$\underline{\int_{a}^{b}}f\le\int f\,d\lambda'\le\overline{\int_{a}^{b}}f$$

(The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iii), Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv)). As $f$ is Riemann integrable, both outer terms equal $\int_{a}^{b}f$ (The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iv), The Riemann Integral on a Closed Interval via Lower and Upper Integrals (v)), so $\int_{a}^{b}f\le\int f\,d\lambda'\le\int_{a}^{b}f$, and

$$\int f\,d\lambda'=\int_{a}^{b}f$$

(Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (viii)).

Proof of (iii). The function $f$ is bounded and Riemann integrable on $[a,b]$ (A Continuous Function on a Closed Interval is Riemann Integrable (i)). It is continuous as a function from the metric space $([a,b],d_{[a,b]})$ to $(\mathbb{R},d_{\mathbb{R}})$, hence $\mathcal{B}([a,b])/\mathcal{B}(\mathbb{R})$-measurable, and $\mathcal{B}([a,b])=\mathcal{B}'$ by (i), so $f$ is measurable (The Real Line: Standing Notation and Background for Calculus on Intervals (vi), The Real Line: Standing Notation and Background for Calculus on Intervals (iv), Continuity on a Subset Is Continuity on the Metric Subspace (ii), Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv), Borel Sigma-Algebra of a Metric Space (ii)). The rest is (ii).