Development of the Darboux integral

Overview §

If $D\subset {\mathbb{R}}^n$ is Jordan measurable and $f:D\to \mathbb{R}$ is a continuous and bounded function, then the Darboux integral ${\int }_{D}f(\vec{x})d{\text{vol}}_n$ is defined. Though not unambiguously correct like the Lebesgue integral, this gives a “reasonable” notion of integrability that allows one to do calculus.

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Darboux integral in $\mathbb{R}$ §

Definition: Partition

Given a bounded set $[a,b]\subset \mathbb{R}$, a partition $P=\{x_0,\dots ,x_n\}$ of $[a,b]$ is the collection of intervals made from a finite list of “sampling points” with $a=x_0<\cdots <x_n=b$.

A refinement of this partition is made by adding sampling points to generate new intervals. Two partitions of $P,P^{\prime }$ need not be comparable, but one can use their common refinement

$$P\cup P^{\prime }=\{x_0,\dots ,x_n,x_0^{\prime },\dots ,x_m^{\prime }\}.$$

Given a partition $P=\{x_0,\dots ,x_n\}$ of $[a,b]\subset \mathbb{R}$, the upper and lower sums are motivated by the fact that for any $k$ and $t\in [x_{k-1},x_k]$:

• $[x_{k-1},x_k]\times [0,\sup f(t)]$ is the largest box containing the graph of $f$;
• $[x_{k-1},x_k]\times [0,\inf f(t)]$ is the largest box contained under the graph of $f$.

Definition: Darboux sums and integrals, Darboux integrable in a single variable

Let $f:[a,b]\to \mathbb{R}$ be a function, and let $P=\{x_0,\dots ,x_n\}$ be a partition of $[a,b]$. Then the upper and lower Darboux sums are defined by

$$U(f,P)=\sum _{k=1}^n(x_k-x_{k-1})\underset{t\in [x_{k-1},x_k]}{\sup }f(t)$$ $$L(f,P)=\sum _{k=1}^n(x_k-x_{k-1})\underset{t\in [x_{k-1},x_k]}{\inf }f(t),$$

respectively. Further, the upper and lower Darboux integrals are given by taking the infimum and supremum of all partitions of $[a,b]$, respectively:

$$U_f=\underline{{\int }_a^b}f=\inf \{U(f,P)|P\text{ is a partition of }[a,b]\}$$ $$L_f=\overline{{\int }_a^b}f=\sup \{L(f,P)|P\text{ is a partition of }[a,b]\}.$$

If $U_f=L_f$, then $f$ is said to be Darboux integrable.

Intuitively, if $P’$ is a direct refinement of a partition $P$ such that the intervals or rectangle widths are smaller, then $U(f,P’)\le U(f,P)$ and $L(f,P’)\ge L(f,P)$.

Theorem (Rudin 6.5): The upper Darboux sum is greater than or equal to the lower Darboux sum

Let $f:[a,b]\to \mathbb{R}$ be bounded. Then

$$L_f=\underset{\text{partitions }P}{\sup }L(f,P)\le U_f=\underset{\text{partitions }P}{\inf }U(f,P).$$

Proof from Modern Analysis I.

Let $P_1,P_2$ be two partitions of $[a,b]$. These partitions are not directly comparable, so we instead use their common refinement $Q=P_1\cup P_2$. (🔺 Show that the lower Darboux sum of a refinement is $\ge$ the original lower sum, and the upper sum of the refinement is $\le$ the original upper sum.) Then $L(f,P_1)\le L(f,Q)$ and $U(f,Q)\le U(f,P_2)$, and $L(f,Q)\le U(f,Q)$ by definition, so all together we have

Since $P_1,P_2$ are arbitrary and we have for any two partitions

$$L(f,P_1)\le L_f{U}_f\le U(f,P_2),$$

it follows that $L(f,P_1)\le U(f,P_2)$. Then taking the supremum over all $P_1$ and infimum over all $P_2$ gives proves the stated theorem. $\square$

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Darboux integral in higher dimensions §

Definition: Darboux integrable in multiple variables

A function $f:[a_1,b_1]\times \cdots [a_n,b_n]\to \mathbb{R}$ is Darboux integrable if the greatest lower bound of the sum of “overestimate boxes” $U_f$ and the least upper bound of the sum of “underestimate boxes” $L_f$ are the same. In this case, we set its integral to be In this case, we set its integral to be

$$U_f=L_f={\int }_{B}f(\vec{x})d{\text{vol}}_n$$

where $B$ is the box of the domain.

Definition: Characteristic function

If $D\subset {\mathbb{R}}^n$ is a subset of Euclidian space, its characteristic function ${\chi }_D:{\mathbb{R}}^n\to \mathbb{R}$ is

$${\chi }_D(\vec{x})=\{\begin{array}{ll}1& \vec{x}\in D\\ 0& \vec{x}\notin D\end{array}.$$

Definition: Jordan measurable

A subset of Euclidian space $D\subset {\mathbb{R}}^n$ is Jordan measurable if its characteristic function ${\chi }_D$ is integrable. In this case, we set

$${\text{vol}}_n(D)={\int }_B{\chi }_{D}d{\text{vol}}_n$$

for some box $B$ containing $D$.

Remark. What does Jordan measurable actually mean? If ${\chi }_D$ is integrable, we can compute upper and lower sums for ${\chi }_D$, which are over and underestimate approximations using boxes. - Then $D$ is Jordan measurable if for all $ϵ>0$, the inner and outer boxes are at least $ϵ$-close. - In other words, the domain is Jordan measurable if its “boundary” has volume (i.e., Jordan measure) $0$!

Definition: Integration on Jordan measurable sets

If $D\subset {\mathbb{R}}^n$ is a bounded subset (that is, it is contained in a “box” with $D\subset [-c,c{]}^n$ for some $c>0$), and $f:D\to \mathbb{R}$ is a function, we say $f$ is integrable on $D$ if $f{\chi }_D$ is integrable over the box $[-c,c{]}^n$, with integral

$${\int }_{D}f(\vec{x})d{\text{vol}}_n={\int }_{[-c,c{]}^n}f(\vec{x}){\chi }_D(\vec{x})d{\text{vol}}_n.$$

%% ## Darboux integration in higher dimensions

• (Definition) A function $f:[a_1,b_1]\times \cdots [a_n,b_n]\to \mathbb{R}$ is Darboux integrable if the greatest lower bound of the sum of “overestimate boxes” $U_f$ and the least upper bound of the sum of “underestimate boxes” $L_f$ are the same. In this case, weset its integral to be $U_f=L_f={\int }_{B}f(\vec{x})d{\text{vol}}_n$ where $B$ is the box of the domain.

• (Definition) If $D\subset {\mathbb{R}}^n$ is a subset of Euclidian space, its characteristic function ${\chi }_D:{\mathbb{R}}^n\to \mathbb{R}$ is ${\chi }_D(\vec{x})=\{\begin{array}{ll}1& \vec{x}\in D\\ 0& \vec{x}\notin D\end{array}.$

• (Definition) If $D\subset {\mathbb{R}}^n$ is a bounded subset (that is, it is contained in a “box” with $D\subset [-c,c{]}^n$ for some $c>0$), and $f:D\to \mathbb{R}$ is a function, we say $f$ is integrable on $D$ if $f{\chi }_D$ is integrable over the box $[-c,c{]}^n$, with integral ${\int }_{D}f(\vec{x})d{\text{vol}}_n={\int }_{[-c,c{]}^n}f(\vec{x}){\chi }_D(\vec{x})d{\text{vol}}_n.$

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Jordan measurability §

• (Definition) A subset of Euclidian space $D\subset {\mathbb{R}}^n$ is Jordan measurable if its characteristic function ${\chi }_D$ is integrable. In this case, we set ${\text{vol}}_n(D)={\int }_B{\chi }_{D}d{\text{vol}}_n$ for some box $B$ containing $D$.
• What does Jordan measurable actually mean?
  • If ${\chi }_D$ is integrable, we can compute upper and lower sums for ${\chi }_D$, which are over and underestimate approximations using boxes.
  • Then $D$ is Jordan measurable if for all $ϵ>0$, the inner and outer boxes are at least $ϵ$-close.
  • In other words, the domain is Jordan measurable if its “boundary” has volume (i.e., Jordan measure) $0$!
• (Proposition 16.1) If $D_1,D_2\subset {\mathbb{R}}^n$ are Jordan measurable, then so are their intersection $D_1\cap D_2$ and union $D_1\cup D_2$, and their volumes can be computed by ${\text{vol}}_n(D_1)+{\text{vol}}_n(D_2)-{\text{vol}}_n(D_1\cap D_2)={\text{vol}}_n(D_1\cup D_2).$
• (Proposition 16.2) The region bounded by continuous bounded functions is Jordan measurable. If $D\subset B\subset {\mathbb{R}}^{n-1}$ is Jordan measurable and $f,g:D\to \mathbb{R}$ are continuous and bounded, $D^{\prime }=\{\vec{x}\in {\mathbb{R}}^n\Vert f(\vec{x})\le x_j\le g(\vec{x})\}$ is Jordan measurable, where $x_j$ is the coordinate omitted in $D\subset {\mathbb{R}}^{n-1}$.
• (Theorem 16.3) If $D\subset {\mathbb{R}}^n$ is Jordan measurable and $f:D\to \mathbb{R}$ is continuous and bounded, then $f$ is Darboux integrable on $D$. %%

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## Darboux integration in higher dimensions

- *(Definition)* A function $f: [a_1,b_1] \times \cdots [a_n,b_n] \to \mathbb R$ is **Darboux integrable** if the *greatest lower bound* of the sum of "overestimate boxes" $U_f$ and the *least upper bound* of the sum of "underestimate boxes" $L_f$ are the same. In this case, weset its integral to be $$ U_f = L_f = \int_B f(\vec x) d\text{vol}_n $$ where $B$ is the box of the domain.

- *(Definition)* If $D \subset \mathbb R^n$ is a subset of Euclidian space, its **characteristic function** $\chi_D : \mathbb R^n \to \mathbb R$ is $$ \chi_D(\vec x) = \begin{cases} 1 & \vec x \in D \\ 0 & \vec x \notin D\end{cases}. $$
- *(Definition)* If $D \subset \mathbb R^n$ is a **bounded subset** (that is, it is contained in a "box" with $D \subset [-c, c]^n$ for some $c > 0$), and $f : D \to \mathbb R$ is a function, we say $f$ is **integrable** on $D$ if $f\chi_D$ is **integrable** over the box $[-c, c]^n$, with integral $$ \int_D f(\vec x) d\text{vol}_n = \int_{[-c, c]^n} f(\vec x) \chi_D(\vec x) d \text{vol}_n. $$
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## Jordan measurability

- *(Definition)* A subset of Euclidian space $D \subset \mathbb R^n$ is **Jordan measurable** if its **characteristic function** $\chi_D$ is **integrable**. In this case, we set $$ \text{vol}_n(D) = \int_B \chi_D d\text{vol}_n $$ for some box $B$ containing $D$.
- What does **Jordan measurable** actually mean?
	- If $\chi_D$ is **integrable**, we can compute upper and lower sums for $\chi_D$, which are over and underestimate approximations using boxes.
	- Then $D$ is **Jordan measurable** if for all $\epsilon > 0$, the inner and outer boxes are at least $\epsilon$-close.
	- In other words, the domain is **Jordan measurable** if its "[[Fubini's theorem and change of variables|boundary]]" has volume (i.e., **Jordan measure**) $0$!
- *(Proposition 16.1)* If $D_1, D_2 \subset \mathbb R^n$ are **Jordan measurable**, then so are their intersection $D_1 \cap D_2$ and union $D_1 \cup D_2$, and their volumes can be computed by $$ \text{vol}_n(D_1) + \text{vol}_n(D_2) - \text{vol}_n(D_1 \cap D_2) = \text{vol}_n(D_1 \cup D_2). $$
- *(Proposition 16.2)* **The region bounded by continuous bounded functions is Jordan measurable.** If $D \subset B \subset \mathbb R^{n-1}$ is **Jordan measurable** and $f,g : D \to \mathbb R$ are **continuous** and **bounded**, $$ D' = \{\vec x \in \mathbb R^n \| \ f(\vec x) \leq x_j \leq g(\vec x) \} $$ is **Jordan measurable**, where $x_j$ is the coordinate omitted in $D \subset \mathbb R^{n-1}$. 
- *(Theorem 16.3)* If $D \subset \mathbb R^n$ is **Jordan measurable** and $f: D \to \mathbb R$ is [[Continuity|continuous]] and **bounded**, then $f$ is **Darboux integrable** on $D$.