Darboux integral

The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real-valued function $f$ on the interval $[a,b].$ 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:

Image:Darboux.svg

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

:

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

:$P\; =\; (x\_0,\; \backslash ldots,\; x\_n)$

be a partition of $[a,b]$. Let

:$\backslash begin\{align\}$

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{align}

The upper Darboux sum of $f$ with respect to $P$ is

:$U\_\{f,\; P\}\; =\; \backslash sum\_\{i=1\}^n\; (x\_\{i\}-x\_\{i-1\})\; M\_i.\; \backslash ,\backslash !$

The lower Darboux sum of $f$ with respect to $P$ is

:$L\_\{f,\; P\}\; =\; \backslash sum\_\{i=1\}^n\; (x\_\{i\}-x\_\{i-1\})\; m\_i.\; \backslash ,\backslash !$

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

The upper Darboux integral of f is

:$U\_f\; =\; \backslash inf\backslash \{U\_\{f,P\}\; \backslash colon\; P\; \backslash text\{\; is\; a\; partition\; of\; \}\; [a,b]\backslash \}.$

The lower Darboux integral of f is

:$L\_f\; =\; \backslash sup\backslash \{L\_\{f,P\}\; \backslash colon\; P\; \backslash text\{\; is\; a\; partition\; of\; \}\; [a,b]\backslash \}.$

In some literature, an integral symbol with an underline and overline represent the lower and upper Darboux integrals respectively:

:$\backslash begin\{align\}$

&{} L_f \equiv \underline{\int_{a}^{b}} f(x) \, \mathrm{d}x, \\

&{} U_f \equiv \overline{\int_{a}^{b}} f(x) \, \mathrm{d}x,

\end{align}

and like Darboux sums they are sometimes simply called the lower and upper integrals.

If U_f = L_f, then we call the common value the Darboux integral.Wolfram MathWorld We also say that f is Darboux-integrable or simply integrable and set

:$\backslash int\_a^b\; \{f(t)\backslash ,dt\}\; =\; U\_f\; =\; L\_f.$

An equivalent and sometimes useful criterion for the integrability of f is to show that for every ε > 0 there exists a partition P_ε of [a, b] such thatSpivak 2008, chapter 13.

:

• 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).
• :$(b-a)\backslash inf\_\{x\; \backslash in\; [a,b]\}\; f(x)\; \backslash leq\; L\_\{f,P\}\; \backslash leq\; U\_\{f,P\}\; \backslash leq\; (b-a)\backslash sup\_\{x\; \backslash in\; [a,b]\}\; f(x)$
• The lower and upper Darboux integrals satisfy
• :$\backslash underline\{\backslash int\_\{a\}^\{b\}\}\; f(x)\; \backslash ,\; dx\; \backslash leq\; \backslash overline\{\backslash int\_\{a\}^\{b\}\}\; f(x)\; \backslash ,\; dx$
• Given any c in (a, b)
• :$\backslash begin\{align\}$

\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{align}

• 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:
• :$\backslash begin\{align\}$

\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{align}

• For a constant c ≥ 0 we have
• :$\backslash begin\{align\}$

\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{align}

• For a constant c ≤ 0 we have
• :$\backslash begin\{align\}$

\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{align}

• Consider the function
• :$\backslash begin\{align\}$

&{} F : [a, b] \to \R \\

&{} F(x) = \underline{\int_{a}^{x}} f(t) \, dt,

\end{align}

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

Suppose we want to show that the function $f(x)=x$ is Darboux-integrable on the interval $[0,1]$ and determine its value. To do this we partition $[0,1]$ into $n$ equally sized subintervals each of length $1/n$. We denote a partition of $n$ equally sized subintervals as $P\_n$.

Now since $f(x)=x$ is strictly increasing on $[0,1]$, the infimum on any particular subinterval is given by its starting point. Likewise the supremum on any particular subinterval is given by its end point. The starting point of the $k$-th subinterval in $P\_n$ is $(k-1)/n$ and the end point is $k/n$. Thus the lower Darboux sum on a partition $P\_n$ is given by

:$\backslash begin\{align\}$

L_{f,P_n} &= \sum_{k = 1}^{n} f(x_{k-1})(x_{k} - x_{k-1}) \\

&= \sum_{k = 1}^{n} \frac{k-1}{n} \cdot \frac{1}{n} \\

&= \frac{1}{n^2} \sum_{k = 1}^{n} [k-1] \\

&= \frac{1}{n^2}\left[ \frac{(n-1)n}{2} \right] \\

&= \frac{1}{2} - \frac{1}{2n}

\end{align}

similarly, the upper Darboux sum is given by

:$\backslash begin\{align\}$

U_{f,P_n} &= \sum_{k = 1}^{n} f(x_{k})(x_{k} - x_{k-1}) \\

&= \sum_{k = 1}^{n} \frac{k}{n} \cdot \frac{1}{n} \\

&= \frac{1}{n^2} \sum_{k = 1}^{n} k \\

&= \frac{1}{n^2}\left[ \frac{(n+1)n}{2} \right] \\

&= \frac{1}{2} + \frac{1}{2n}

\end{align}

Since

:$U\_\{f,P\_n\}\; -\; L\_\{f,P\_n\}\; =\; \backslash frac\{1\}\{n\}$

Thus for given any $\backslash varepsilon>0$, we have that any partition $P\_n$ with $n\; >\; \backslash frac\{1\}\{\backslash varepsilon\}$ satisfies

:

which shows that $f$ is Darboux integrable. To find the value of the integral note that

:$\backslash int\_\{0\}^\{1\}f(x)\; \backslash ,\; dx$

= \lim_{n \to \infty} U_{f,P_n}

= \lim_{n \to \infty} L_{f,P_n}

= \frac{1}{2}

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Suppose we have the Dirichlet function $f:\backslash R\; \backslash to\; [0,1]$ defined as

:$\backslash begin\{align\}$

f(x) &=

\begin{cases}

1 & \text{if }x\text{ is rational} \\

0 & \text{if }x\text{ is irrational}

\end{cases}

\end{align}

Since the rational and irrational numbers are both dense subsets of $\backslash mathbb\{R\}$, it follows that $f$ takes on the value of 0 and 1 on every subinterval of any partition. Thus for any partition $P$ we have

:$\backslash begin\{align\}$

L_{f,P} &=\sum_{k = 1}^{n}(x_{k} - x_{k-1})\inf_{x \in [x_{k-1},x_{k}]}f = 0 \\

U_{f,P} &=\sum_{k = 1}^{n}(x_{k} - x_{k-1}) \sup_{x \in [x_{k-1},x_{k}]}f = 1

\end{align}

from which we can see that the lower and upper Darboux integrals are unequal.

Image:Darboux refinement.svg

A refinement of the partition $x\_0,\; \backslash ldots,\; x\_n$ is a partition $y\_0,\; \backslash ldots,\; y\_m$ such that for all i = 0, …, n there is an integer r(i) such that

:$x\_\{i\}\; =\; y\_\{r(i)\}\; .$

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

If $P\text{'}\; =\; (y\_0,\backslash ldots,y\_m)$ is a refinement of $P\; =\; (x\_0,\backslash ldots,x\_n)\; ,$ then

:$U\_\{f,\; P\}\; \backslash ge\; U\_\{f,\; P\text{'}\}$

and

:$L\_\{f,\; P\}\; \backslash le\; L\_\{f,\; P\text{'}\}.$

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

:$L\_\{f,\; P\_1\}\; \backslash le\; U\_\{f,\; P\_2\},$

and it follows that

:$L\_f\; \backslash le\; U\_f\; .$

Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if $P\; =\; (x\_0,\backslash ldots,x\_n)$ and $T\; =\; (t\_1,\backslash ldots,t\_n)$ together make a tagged partition

:$x\_0\; \backslash le\; t\_1\; \backslash le\; x\_1\backslash le\; \backslash cdots\; \backslash le\; x\_\{n-1\}\; \backslash le\; t\_n\; \backslash le\; x\_n$

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

:$L\_\{f,\; P\}\; \backslash le\; R\_\{f,\; P\}\; \backslash le\; U\_\{f,\; P\}.$

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.

:

• Regulated integral
• Lebesgue integration
• Minimum bounding rectangle

{{Reflist}}

• {{cite web

| url = http://mathworld.wolfram.com/DarbouxIntegral.html

| title = Darboux Integral

| access-date = 2013-01-08

| work=Wolfram MathWorld

}}

• [http://www.encyclopediaofmath.org/index.php/Darboux_integral Darboux integral at Encyclopaedia of Mathematics]
• {{springer|title=Darboux sum|id=p/d030160}}
• {{Citation

| last = Spivak

| first = Michael

| title = Calculus

| publisher = Publish or Perish

| year = 2008

| edition = 4

| isbn = 978-0914098911

| url-access = registration

| url = https://archive.org/details/calculus4thediti00mich

}}

• {{cite web

| url=https://www.youtube.com/watch?v=W7Ki6y9jAXc

| title=Equivalence of Darboux and Riemann integral

}}{{cbignore}}{{Dead Youtube links|date=February 2022}}

{{Integral}}

{{DEFAULTSORT:Darboux Integral}}

Category:Definitions of mathematical integration