Darboux integral

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In real analysis, a branch of mathematics, the Darboux integral is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. Darboux integrals have the advantage of being simpler to define than Riemann integrals. Darboux integrals are named after their discoverer: Gaston Darboux.

Contents

Definition

A partition of an interval [a, b] is a finite sequence

a = x0 < x1 < x2 < ... < xn = b.

Each [xi, xi+1] is called a subinterval of the partition. A refinement of the partition

x0, ..., xn

is a partition

y0, ..., ym

such that for every i with

0≤in,

there is an integer r(i) such that

xi=yr(i).

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

Let f:[a,b]→R be a bounded function, and let

x0, ..., xn

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

<math>M_i = \sup_{x\in[x_i,x_{i+1}]} f(x)<math>
<math>m_i = \inf_{x\in[x_i,x_{i+1}]} f(x)<math>

The upper Darboux sum of f with respect to x0, ..., xn is

<math>U_{f, x_0,\ldots,x_n} = \sum_{i=0}^n M_i (x_{i+1}-x_i)<math>

The lower Darboux sum of f with respect to x0, ..., xn is

<math>L_{f, x_0,\ldots,x_n} = \sum_{i=0}^n m_i (x_{i+1}-x_i)<math>

The upper Darboux integral of f is

<math>U_f = \inf_{x_0,\ldots,x_n} U_{f, x_0,\ldots,x_n}<math>

The lower Darboux integral of f is

<math>L_f = \sup_{x_0,\ldots,x_n} L_{f, x_0,\ldots,x_n}<math>

If Uf = Lf, then we say that f is Darboux-integrable and set ∫f to be the common value of the upper and lower Darboux integrals.

Facts about the Darboux integral

If

y0, ..., ym

is a refinement of

x0, ..., xn,

then

<math>U_{f, x_0,\ldots,x_n} \ge U_{f, y_0,\ldots,y_m}<math>

and

<math>L_{f, x_0,\ldots,x_n} \le L_{f, y_0,\ldots,y_m}<math>

If

x0, ..., xn and
y0, ..., ym

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

<math>L_{f, x_0,\ldots,x_n} \le U_{f, y_0,\ldots,y_m}<math>.

It follows that

LfUf.

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

x0, ..., xn

and

t0,...,tm-1

together make a tagged partition (as in the definition of the Riemann integral), and if the Riemann sum of f corresponding to xn and

t0,...,tm-1

is R, then

<math>L_{f, x_0,\ldots,x_n} \le R \le U_{f, x_0,\ldots,x_n}<math>.

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

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