Darboux Integral

--

In .

DEFINITION

A Partition Of An Interval

{Link without Title} is a finite sequence

:

a = x_0 < x_1 < \cdots < x_n = b

Each

{Link without Title} is called a ''subinterval'' of the partition. A ''refinement'' of the partition

:

x_0,\ldots,x_n

is a partition

:

y_0, \ldots, y_m

such that for every

i

with

:

0 \le i \le n

there is an integer

r(i)

such that

:

x_i = y_{r(i)}

In other words, to make a refinement, one cuts the subintervals into smaller pieces and does not remove any cuts.

Let

f :

{Link without Title} ightarrow \mathbb{R} be a bounded function, and let

:

P : x_0, \ldots, x_n

be a partition of

{Link without Title} . Let:

:

M_i = \sup_{x\in

{Link without Title} } f(x)
:

m_i = \inf_{x\in

{Link without Title} } f(x)

The upper Darboux sum of

f

with respect to

P

is

:

U_{f, P} = \sum_{i=1}^n M_i (x_{i}-x_{i-1})

The lower Darboux sum of

f

with respect to

P

is

:

L_{f, P} = \sum_{i=1}^n m_i (x_{i}-x_{i-1})

The upper Darboux integral of

f

is

:

U_f = \inf\{U_{f,P} : 	extrm{P\, is\, a\, partition\, of\,}

{Link without Title} \}

The lower Darboux integral of

f

is

:

L_f = \sup\{L_{f,P} : 	extrm{P\, is\, a\, partition\, of\,}

{Link without Title} \}

If

U_f=L_f

, then we say that

f

is ''Darboux-integrable'' and set

\int_a^b{f(t)\,dt}

to be the common value of the upper and lower Darboux integrals.

FACTS ABOUT THE DARBOUX INTEGRAL

If

:

P' : y_0,\ldots,y_m

is a refinement of

:

P : x_0,\ldots,x_n

,

then

:

U_{f, P} \ge U_{f, P'}

and

:

L_{f, P} \le L_{f, P'}

P_1,P_2

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

:

L_{f, P_1} \le U_{f, P_2}

.

It follows that

:

L_f \le U_f

Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if

:

P : x_0,\ldots,x_n

and

:

T : t_0,\ldots,t_{n-1}

together make a tagged partition (as in the definition of the Riemann Integral ), and if the Riemann sum of

f

corresponding to

P

and

T

R

, then

:

L_{f, P} \le R \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. It is not hard to see that there is 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.

SEE ALSO

Riemann Integral

Lebesgue Integration

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Information About

Darboux Integral