The Riemann Integral on a Closed Interval via Lower and Upper Integrals

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $a,b\in\mathbb{R}$ with $a<b$, and let $f:[a,b]\to\mathbb{R}$ be bounded: there is $B\in\mathbb{R}$ with $|f(t)|\le B$ for every $t\in[a,b]$. Let $\Pi_{a,b}$ be the set of partitions of $[a,b]$, as in Partitions of a Closed Interval, the Set of Partitions, and Refinements (ii), and for $R\in\Pi_{a,b}$ let $U(f,R)$ and $L(f,R)$ be the upper sum and the lower sum of $f$ with respect to $R$, as in Upper and Lower Darboux Sums of a Bounded Function on a Partition (ii) and Upper and Lower Darboux Sums of a Bounded Function on a Partition (iii).

(i) The set of lower sums of $f$ is

$$\mathcal{L}(f)=\{L(f,R):R\in\Pi_{a,b}\},$$

a set and a subset of $\mathbb{R}$ by Predicative Formulas for Upper and Lower Darboux Sums, and the Maps and Sets of Darboux Sums over All Partitions (ii). By Predicative Formulas for Upper and Lower Darboux Sums, and the Maps and Sets of Darboux Sums over All Partitions (ii) and Partitions of a Closed Interval, the Set of Partitions, and Refinements (ii), for every set $s$: $s\in\mathcal{L}(f)$ if and only if $s=L(f,R)$ for some $R\in\Pi_{a,b}$, that is, for some partition $R$ of $[a,b]$.

(ii) The set of upper sums of $f$ is

$$\mathcal{U}(f)=\{U(f,R):R\in\Pi_{a,b}\},$$

a set and a subset of $\mathbb{R}$ by Predicative Formulas for Upper and Lower Darboux Sums, and the Maps and Sets of Darboux Sums over All Partitions (ii). By Predicative Formulas for Upper and Lower Darboux Sums, and the Maps and Sets of Darboux Sums over All Partitions (ii) and Partitions of a Closed Interval, the Set of Partitions, and Refinements (ii), for every set $s$: $s\in\mathcal{U}(f)$ if and only if $s=U(f,R)$ for some $R\in\Pi_{a,b}$, that is, for some partition $R$ of $[a,b]$.

(iii) The lower integral and the upper integral of $f$ over $[a,b]$ are

$$\underline{\int_{a}^{b}}f=\sup\mathcal{L}(f),\qquad\overline{\int_{a}^{b}}f=\inf\mathcal{U}(f),$$

the supremum and the infimum being formed in $\mathbb{R}$. They exist, and they are the numbers $\lambda_{*}$ and $\lambda^{*}$ of Bounds for Darboux Sums, Monotonicity under Refinement, the Supremum of the Lower Sums and the Infimum of the Upper Sums, and Concatenation of Partitions (iv): by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (i) and (i), the upper bounds of $\mathcal{L}(f)$ in $\mathbb{R}$ are the $\mu\in\mathbb{R}$ with $L(f,R)\le\mu$ for every partition $R$ of $[a,b]$, so, by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order (iv), $\lambda_{*}$ is a supremum of $\mathcal{L}(f)$, and it is the only one; likewise, by (ii), $\lambda^{*}$ is the infimum of $\mathcal{U}(f)$.

(iv) $f$ is Riemann integrable on $[a,b]$, or integrable on $[a,b]$, if $\underline{\int_{a}^{b}}f=\overline{\int_{a}^{b}}f$.

(v) If $f$ is integrable on $[a,b]$, its Riemann integral over $[a,b]$ is the common value

$$\int_{a}^{b}f=\int_{a}^{b}f(t)\,dt=\underline{\int_{a}^{b}}f=\overline{\int_{a}^{b}}f .$$

(vi) Let $E\subseteq\mathbb{R}$, let $g:E\to\mathbb{R}$, and let $u,v\in\mathbb{R}$ with $u<v$ and $[u,v]\subseteq E$. Then $g$ is integrable on $[u,v]$ if the restriction $g|_{[u,v]}:[u,v]\to\mathbb{R}$ is bounded and integrable on $[u,v]$, and in that case $\int_{u}^{v}g=\int_{u}^{v}g(t)\,dt$ denotes $\int_{u}^{v}g|_{[u,v]}$. If $E=[u,v]$ and $g$ is bounded, then $g|_{[u,v]}=g$ by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iii), so this agrees with (iv) and (v).

(vii) With $E$ and $g$ as in (vi), we set $\int_{u}^{u}g=\int_{u}^{u}g(t)\,dt=0$ for every $u\in E$; and for $u,v\in\mathbb{R}$ with $u<v$ and $[u,v]\subseteq E$ such that $g$ is integrable on $[u,v]$, we set $\int_{v}^{u}g=\int_{v}^{u}g(t)\,dt=-\int_{u}^{v}g$.

(viii) With $E$ and $g$ as in (vi), for $u,v\in E$ the integral $\int_{u}^{v}g$ is defined if $u=v$; or $u<v$, $[u,v]\subseteq E$ and $g$ is integrable on $[u,v]$ in the sense of (vi); or $v<u$, $[v,u]\subseteq E$ and $g$ is integrable on $[v,u]$ in the sense of (vi). In each case $\int_{u}^{v}g$ is the real number given by (vi) and (vii).