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].

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