Darboux Integral

Definition

Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable $\iff$ it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.

Darboux Sums

A partition of an interval $[a,b]$ is a finite sequence of values $x_i$ such that $a=x_0<x_1<\cdots <x_n=b$ Each interval $[x_{i-1},x_i]$ is called a sub-interval of the partition.

Let $f:[a,b]\to \mathbb{R}$ be a bounded function, $P=(x_0,\dots ,x_n)$ be a partition of $[a,b]$, $\Delta x_i=x_i-x_{i-1}$ $M_i=\sup \{f(x)|x_{i-1}\le x\le x_i\}$, and $m_i=\inf \{f(x)|x_{i-1}\le x\le x_i\}$

The upper Darboux sum of $f$ with respect to $P$ is $U_{f,P}=\sum _{i=1}^n{M}_i\Delta x_i$

The lower Darboux sum of $f$ with respect to $P$ is $L_{f,P}=\sum _{i=1}^n{m}_i\Delta x_i$

Darboux Integrals

The upper Darboux integral of $f$ is $U_f=\overline{{\int }_a^b}f=\overline{{\int }_a^b}f(x)dx=\inf \{U_{f,P}\}$

The lower Darboux integral of $f$ is $L_f=\underline{{\int }_a^b}f=\underline{{\int }_a^b}f(x)dx=\sup \{L_{f,P}\}$

If $U_f=L_f\iff \overline{{\int }_a^b}f=\underline{{\int }_a^b}f$, then we call the common value the Darboux integral and set ${\int }_a^{b}f(x)dx=U_f=L_f$

We also say that $f$ is Darboux-integrable, simply integrable, or $f\in \mathcal{R}[a,b]$, where $\mathcal{R}[a,b]$ is a set of integrable function on a $[a,b]$

Useful criterion for the integrability of $f$

$f\in \mathcal{R}[a,b]\iff \forall ϵ>0,\exists P,U_{f,P}-L_{f,P}<ϵ$

Properties

Refinement of a partition

When $P$ is a partition and, $P\subset P^{\prime }$ is satisfied, $P^{\prime }$ is a refinement of $P$

If $P^{\prime }$ is a refinement of $P$, then $L_{f,P}\le L_{f,P^{\prime }}\le U_{f,P^{\prime }}\le U_{f,P}$

If $P_1,P_2$ are two partitions of the same interval, then $L_{f,P_1}\le U_{f,P_2}$

and It follows that $L_f\le U_f\iff \underline{{\int }_a^b}f\le \overline{{\int }_a^b}f$

Linearity

${\int }_a^b(\alpha f(x)+\beta g(x))dx=\alpha {\int }_a^{b}f(x)dx+\beta {\int }_a^{b}g(x)dx$ The Darboux Integration is a linear transformation

Additivity

$f\in \mathcal{R}[a,b]\iff \forall c\in (a,b),f\in \mathcal{R}[a,c]\wedge f\in \mathcal{R}[c,b]$, where ${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$

Facts

Transclude of Continuous-Function#^c92817

$f,g\in \mathcal{R}[a,b]\Rightarrow f\cdot g\in \mathcal{R}[a,b]$ Be careful! $\int f\cdot g\ne \int f\int g$