Darboux integral

The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real-valued function ${\displaystyle f}$ on the interval ${\displaystyle [a,b].}$ The Darboux integral exists if and only if the upper and lower integrals are equal. The upper and lower integrals are in turn the infimum and supremum, respectively, of upper and lower (Darboux) sums which over- and underestimate, respectively, the "area under the curve." In particular, for a given partition of the interval of integration, the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum, respectively, of f in each subinterval of the partition. These ideas are made precise below:

Darboux sums

Lower (green) and upper (green plus lavender) Darboux sums for four subintervals

A partition of an interval ${\displaystyle [a,b]}$ is a finite sequence of values ${\displaystyle x_{i}}$ such that

${\displaystyle a=x_{0}<x_{1}<\cdots <x_{n}=b.}$

Each interval ${\displaystyle [x_{i-1},x_{i}]}$ is called a subinterval of the partition. Let ${\displaystyle f:[a,b]\to \mathbb {R} }$ be a bounded function, and let

${\displaystyle P=(x_{0},\ldots ,x_{n})}$

be a partition of ${\displaystyle [a,b]}$. Let

${\displaystyle {\begin{aligned}M_{i}=\sup _{x\in [x_{i-1},x_{i}]}f(x),\\m_{i}=\inf _{x\in [x_{i-1},x_{i}]}f(x).\end{aligned}}}$

The upper Darboux sum of ${\displaystyle f}$ with respect to ${\displaystyle P}$ is

${\displaystyle U_{f,P}=\sum _{i=1}^{n}(x_{i}-x_{i-1})M_{i}.\,\!}$

The lower Darboux sum of ${\displaystyle f}$ with respect to ${\displaystyle P}$ is

${\displaystyle L_{f,P}=\sum _{i=1}^{n}(x_{i}-x_{i-1})m_{i}.\,\!}$

The lower and upper Darboux sums are often called the lower and upper sums.

• For any given partition, the upper Darboux sum is always greater than or equal to the lower Darboux sum. Furthermore, the lower Darboux sum is bounded below by the rectangle of width (b−a) and height inf(f) taken over [a, b]. Likewise, the upper sum is bounded above by the rectangle of width (b−a) and height sup(f).

  ${\displaystyle (b-a)\inf _{x\in [a,b]}f(x)\leq L_{f,P}\leq U_{f,P}\leq (b-a)\sup _{x\in [a,b]}f(x)}$

• The lower and upper Darboux integrals satisfy

  ${\displaystyle {\underline {\int _{a}^{b}}}f(x)\,dx\leq {\overline {\int _{a}^{b}}}f(x)\,dx}$

• Given any c in (a, b)

  ${\displaystyle {\begin{aligned}{\underline {\int _{a}^{b}}}f(x)\,dx&={\underline {\int _{a}^{c}}}f(x)\,dx+{\underline {\int _{c}^{b}}}f(x)\,dx\\[6pt]{\overline {\int _{a}^{b}}}f(x)\,dx&={\overline {\int _{a}^{c}}}f(x)\,dx+{\overline {\int _{c}^{b}}}f(x)\,dx\end{aligned}}}$

• The lower and upper Darboux integrals are not necessarily linear. Suppose that g:[a, b] → R is also a bounded function, then the upper and lower integrals satisfy the following inequalities:

  ${\displaystyle {\begin{aligned}{\underline {\int _{a}^{b}}}f(x)\,dx+{\underline {\int _{a}^{b}}}g(x)\,dx&\leq {\underline {\int _{a}^{b}}}(f(x)+g(x))\,dx\\[6pt]{\overline {\int _{a}^{b}}}f(x)\,dx+{\overline {\int _{a}^{b}}}g(x)\,dx&\geq {\overline {\int _{a}^{b}}}(f(x)+g(x))\,dx\end{aligned}}}$

• For a constant c ≥ 0 we have

  ${\displaystyle {\begin{aligned}{\underline {\int _{a}^{b}}}cf(x)\,dx&=c{\underline {\int _{a}^{b}}}f(x)\,dx\\[6pt]{\overline {\int _{a}^{b}}}cf(x)\,dx&=c{\overline {\int _{a}^{b}}}f(x)\,dx\end{aligned}}}$

• For a constant c ≤ 0 we have

  ${\displaystyle {\begin{aligned}{\underline {\int _{a}^{b}}}cf(x)\,dx&=c{\overline {\int _{a}^{b}}}f(x)\,dx\\[6pt]{\overline {\int _{a}^{b}}}cf(x)\,dx&=c{\underline {\int _{a}^{b}}}f(x)\,dx\end{aligned}}}$

• Consider the function

  ${\displaystyle {\begin{aligned}&{}F:[a,b]\to \mathbb {R} \\&{}F(x)={\underline {\int _{a}^{x}}}f(t)\,dt,\end{aligned}}}$

then F is Lipschitz continuous. An identical result holds if F is defined using an upper Darboux integral.

When passing to a refinement, the lower sum increases and the upper sum decreases.

A refinement of the partition ${\displaystyle x_{0},\ldots ,x_{n}}$ is a partition ${\displaystyle y_{0},\ldots ,y_{m}}$ such that for all i = 0, …, n there is an integer r(i) such that

${\displaystyle x_{i}=y_{r(i)}.}$

In other words, to make a refinement, cut the subintervals into smaller pieces and do not remove any existing cuts.

If ${\displaystyle P'=(y_{0},\ldots ,y_{m})}$ is a refinement of ${\displaystyle P=(x_{0},\ldots ,x_{n}),}$ then

${\displaystyle U_{f,P}\geq U_{f,P'}}$

and

${\displaystyle L_{f,P}\leq L_{f,P'}.}$

If P₁, P₂ are two partitions of the same interval (one need not be a refinement of the other), then

${\displaystyle L_{f,P_{1}}\leq U_{f,P_{2}},}$

and it follows that

${\displaystyle L_{f}\leq U_{f}.}$

Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if ${\displaystyle P=(x_{0},\ldots ,x_{n})}$ and ${\displaystyle T=(t_{1},\ldots ,t_{n})}$ together make a tagged partition

${\displaystyle x_{0}\leq t_{1}\leq x_{1}\leq \cdots \leq x_{n-1}\leq t_{n}\leq x_{n}}$

(as in the definition of the Riemann integral), and if the Riemann sum of ${\displaystyle f}$ is equal to R corresponding to P and T, then

${\displaystyle L_{f,P}\leq R\leq U_{f,P}.}$

From the previous fact, Riemann integrals are at least as strong as Darboux integrals: if the Darboux integral exists, then the upper and lower Darboux sums corresponding to a sufficiently fine partition will be close to the value of the integral, so any Riemann sum over the same partition will also be close to the value of the integral. There is (see below) a tagged partition that comes arbitrarily close to the value of the upper Darboux integral or lower Darboux integral, and consequently, if the Riemann integral exists, then the Darboux integral must exist as well.

Details of finding the tags

For this proof, we shall use superscripts to index ${\displaystyle \left\{P^{(n)}\right\}}$ and variables related to it. Let ${\displaystyle \left\{P^{(n)}\right\}}$ be a sequence of arbitrary partitions of ${\displaystyle [a,b]}$ such that ${\displaystyle \|P_{n}\|\to 0}$, whose tags are to be determined. By the definition of infimum, for any ${\displaystyle \epsilon >0}$, we can always find a ${\displaystyle t_{i}^{(n)}\in \left[x_{i}^{(n)},x_{i+1}^{(n)}\right]}$ such that ${\displaystyle \inf _{x\in \left[x_{i}^{(n)},x_{i+1}^{(n)}\right]}f(x)\geq f(t_{i}^{(n)})-\epsilon .}$ Thus, ${\displaystyle {\begin{aligned}\sum _{i=0}^{N^{(n)}-1}f(t_{i})(x_{i+1}^{(n)}-x_{i}^{(n)})&\leq &\sum _{i=0}^{N^{(n)}-1}\left(\inf _{x\in \left[x_{i}^{(n)},x_{i+1}^{(n)}\right]}f(x)+\epsilon \right)(x_{i+1}^{(n)}-x_{i}^{(n)})&\ \ \ \\&=&\sum _{i=0}^{N^{(n)}-1}\inf _{x\in \left[x_{i}^{(n)},x_{i+1}^{(n)}\right]}f(x)(x_{i+1}^{(n)}-x_{i}^{(n)})+\sum _{i=0}^{N-1}\epsilon (x_{i+1}^{(n)}-x_{i}^{(n)})\\&=&\sum _{i=0}^{N^{(n)}-1}\inf _{x\in \left[x_{i}^{(n)},x_{i+1}^{(n)}\right]}f(x)(x_{i+1}^{(n)}-x_{i}^{(n)})+\epsilon (b-a).\end{aligned}}}$ Let ${\displaystyle \epsilon =1/n(b-a)}$, we have ${\displaystyle {\begin{aligned}\sum _{i=0}^{N^{(n)}-1}f(t_{i})(x_{i+1}^{(n)}-x_{i}^{(n)})&\leq &\sum _{i=0}^{N^{(n)}-1}\inf _{x\in \left[x_{i}^{(n)},x_{i+1}^{(n)}\right]}f(x)(x_{i+1}^{(n)}-x_{i}^{(n)})+{\frac {1}{n}}\\&=&L_{f,P^{(n)}}+{\frac {1}{n}}\end{aligned}}}$ Taking limits of both sides, ${\displaystyle {\begin{aligned}R_{f}=\sum _{i=0}^{N^{(n)}-1}f(t_{i})(x_{i+1}^{(n)}-x_{i}^{(n)})\leq \lim _{n\to \infty }L_{f,P^{(n)}}+\lim _{n\to \infty }{\frac {1}{n}}=\lim _{n\to \infty }L_{f,P^{(n)}}.\end{aligned}}}$ Similarly, (with a different sequences of tags) ${\displaystyle {\begin{aligned}R_{f}\geq \lim _{n\to \infty }U_{f,P^{(n)}}.\end{aligned}}}$ Thus, we have ${\displaystyle R_{f}\leq \lim _{n\to \infty }L_{f,P^{(n)}}\leq \lim _{n\to \infty }U_{f,P^{(n)}}\leq R_{f},}$ which means that the Darboux integral exist and equals ${\displaystyle R_{f}}$.

• Regulated integral
• Lebesgue integration
• Minimum bounding rectangle

• "Darboux Integral". Wolfram MathWorld. Retrieved 2013-01-08.
• Darboux integral at Encyclopaedia of Mathematics
• "Darboux sum", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Spivak, Michael (2008), Calculus (4 ed.), Publish or Perish, ISBN 978-0914098911
• "Equivalence of Darboux and Riemann integral".