Darboux sum

A sum of special type. Let a real function $f$ be defined and bounded on a segment $[ a , b ]$, let $\tau = {\{ x _ {i} \} } _ {i=} 0 ^ {k}$ be a decomposition of $[ a , b ]$:

$$a = x _ {0} < x _ {1} < \dots < x _ {k} = b ,$$

and set

$$m _ {i} = \ \inf _ {x _ {i-} 1 \leq x \leq x _ {i} } f( x),\ \ M _ {i} = \sup _ {x _ {i-} 1 \leq x \leq x _ {i} } f ( x) ,$$

$$\Delta x _ {i} = x _ {i} - x _ {i-} 1 ,\ i = 1 \dots k .$$

The sums

$$s _ \tau = \sum _ {i = 1 } ^ { k } m _ {i} \Delta x _ {i} \ \textrm{ and } \ \ S _ \tau = \sum _ {i = 1 } ^ { k } M _ {i} \Delta x _ {i}$$

are known, respectively, as the lower and upper Darboux sums. For any two decompositions $\tau$ and $\tau ^ \prime$ of $[ a , b ]$ the inequality $s _ \tau \leq S _ {\tau ^ \prime }$ is valid, i.e. any lower Darboux sum is no larger than an upper. If

$$\sigma _ \tau = \sum _ {i = 1 } ^ { k } f ( \xi _ {i} ) \Delta x _ {i} , \ \xi _ {i} \in [ x _ {i-} 1 , x _ {i} ] ,$$

is a Riemann sum, then

$$s _ \tau = \inf _ {\xi _ {1} \dots \xi _ {k} } \sigma _ \tau ,\ S _ \tau = \sup _ {\xi _ {1} \dots \xi _ {k} } \sigma _ \tau .$$

The geometric meaning of the lower and upper Darboux sums is that they are equal to the planar areas of stepped figures consisting of rectangles whose base widths are $\Delta x _ {i}$ and with respective heights $m _ {i}$ and $M _ {i}$( see Fig.) if $f \geq 0$. These figures approximate, from the inside and outside, the curvilinear trapezium formed by the graph of $f$, the abscissa axis and the rectilinear segments $x = a$ and $x= b$( which may degenerate into points).

Figure: d030160a

The numbers

$$\tag{1 } I _ {*} = \sup _ \tau s _ \tau ,\ \ I ^ {*} = \inf _ \tau S _ \tau$$

are called, respectively, the lower and the upper Darboux integrals of $f$. They are the limits of the lower and the upper Darboux sums:

$$I _ {*} = \lim\limits _ {\delta _ \tau \rightarrow 0 } s _ \tau ,\ \ I ^ {*} = \lim\limits _ {\delta _ \tau \rightarrow 0 } S _ \tau ,$$

where

$$\delta _ \tau = \max _ {i = 1 \dots k } \Delta x _ {i}$$

is the fineness (mesh) of the decomposition $\tau$. The condition

$$\tag{2 } I _ {*} = I ^ {*}$$

is necessary and sufficient for a function $f$ to be Riemann integrable on the segment $[ a , b ]$. Here, if condition (2) is met, the value of the lower and the upper Darboux integrals becomes identical with the Riemann integral

$$\int\limits _ { a } ^ { b } f ( x) dx .$$

With the aid of Darboux sums, condition (2) may be formulated in the following equivalent form: For each $\epsilon > 0$ there exists a decomposition $\tau$ such that

$$S _ \tau - s _ \tau < \epsilon .$$

The condition

$$\lim\limits _ {\delta _ \tau \rightarrow 0 } ( S _ \tau - s _ \tau ) = 0$$

is also necessary and sufficient for the Riemann integrability of $f$ on $[ a , b ]$. Here

$$S _ \tau - s _ \tau = \sum _ {i = 1 } ^ { k } \omega _ {i} ( f ) \Delta x _ {i} ,$$

where $\omega _ {i} ( f )$ is the oscillation (cf. Oscillation of a function) of $f$ on

$$[ x _ {i-} 1 , x _ {i} ] ,\ i = 1 \dots k .$$

The concept of lower and upper Darboux sums may be generalized to the case of functions of several variables which are measurable in the sense of some positive measure $\mu$. Let $E$ be a measurable (for example, Jordan or Lebesgue) subset of the $n$- dimensional space, $n = 1 , 2 \dots$ and suppose $\mu ( E)$ is finite. Let $\tau = \{ E _ {i} \} _ {i=} 1 ^ {k}$ be a decomposition of $E$, i.e. a system of measurable subsets of $E$ such that

$$\tag{3 } \cup _ {i= 1 } ^ { k } E _ {i} = E ,$$

$$\tag{4 } \mu ( E _ {i} \cap E _ {j } ) = 0 \ \textrm{ if } i \neq j .$$

Let a function $f$ be bounded on $E$ and let

$$\tag{5 } m _ {i} = \inf _ {x \in E _ {i} } f ( x), \ M _ {i} = \sup _ {x \in E _ {i} } f ( x) ,\ i = 1 \dots k .$$

The sums

$$\tag{6 } s _ \tau = \sum _ {i = 1 } ^ { k } m _ {i} \mu ( E _ {i} ) ,\ \ S _ \tau = \sum _ {i = 1 } ^ { k } M _ {i} \mu ( E _ {i} )$$

are also said to be, respectively, lower and upper Darboux sums. The lower $I _ {*}$ and the upper $I ^ {*}$ integrals are defined by formulas (1). For Jordan measure, their equality is a sufficient and necessary condition for the function to be Riemann integrable, and their common value coincides with the Riemann integral. For Lebesgue measure, on the other hand, the equality

$$I _ {*} = I ^ {*} = \int\limits _ { E } f( x) dx.$$

is always valid for bounded Lebesgue-measurable functions.

In general, if $\mu$ is a complete $\sigma$- additive bounded measure, defined on a $\sigma$- algebra $\mathfrak S _ \mu$, if $f$ is a bounded measurable real-valued function on $E$, if $\tau = \{ E _ {i} \} _ {i= 1 } ^ {k}$ is a decomposition of a set $E \in \mathfrak S _ \mu$ into $\mu$- measurable sets $E _ {i}$ which satisfy the conditions (3) and (4), and if the Darboux sums $s _ \tau$ and $S _ \tau$ are defined by formulas (5) and (6), while the integrals $I _ {*}$ and $I ^ {*}$ are defined by the formulas (1), in which $\mu$ is always understood to mean the measure under consideration, then

$$I _ {*} = I ^ {*} = \int\limits _ { E } f( x) d \mu .$$

A generalization of the Darboux sums to unbounded $\mu$- measurable functions $f$ defined on sets $E \in \mathfrak S _ \mu$ are the series (if they are absolutely convergent)

$$\tag{7 } s _ \tau = \sum _ { i= } 1 ^ \infty m _ {i} \mu ( E _ {i} ) ,\ \ S _ \tau = \sum _ { i= } 1 ^ \infty M _ {i} \mu ( E _ {i} )$$

where $\tau = \{ E _ {i} \} _ {i=} 1 ^ \infty$ is a decomposition of $E \in \mathfrak S _ \mu$( this decomposition consists, generally speaking, of an infinite number of $\mu$- measurable sets $E _ {i}$ which satisfy condition (4) and are, of course, such that $\cup _ {i=} 1 ^ \infty E _ {i} = E$), while $m _ {i}$ and $M _ {i}$ are defined by (5). In (7) (as in (6) above) it is assumed that $\infty \cdot 0 = 0 \cdot \infty = 0$. If $I _ {*}$ and $I ^ {*}$ are again defined according to (1) and $s _ \tau$ and $S _ \tau$ are now defined in the sense of (7) and exist for each $\tau$, then $I _ {*} = I ^ {*}$. If the value $I = I _ {*} = I ^ {*}$ is finite, then $f$ is integrable with respect to $\mu$ and $I = \int _ {E} f ( x) d \mu$.

Named after G. Darboux [1].

References

[1]	G. Darboux, Ann. Sci. Ecole Norm. Sup. Sér. 2 , 4 (1875) pp. 57–112
[2]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[3]	L.D. Kudryavtsev, "Mathematical analysis" , 1–2 , Moscow (1973) (In Russian)
[4]	S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)

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