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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. Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Mathematics is the study of quantity, structure, space and change. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... Jean Gaston Darboux (August 14, 1842, Nîmes – February 23, 1917, Paris) was a French mathematician. ...

Definition

A partition of an interval [a,b] is a finite sequence In mathematics, a partition of an interval [a, b] on the real line is a finite sequence of the form a = x0 < x1 < x2 < ... < xn = b. ...

Each [x_{i − 1},x_i] is called a subinterval of the partition. A refinement of the partition

is a partition

such that for every i with

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 be a bounded function, and let

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

The upper Darboux sum of f with respect to P is

The lower Darboux sum of f with respect to P is

The upper Darboux integral of f is

The lower Darboux integral of f is

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

Facts about the Darboux integral

If

y₀, ..., y_m

is a refinement of

x₀, ..., x_n,

then

and

If

x₀, ..., x_n and

y₀, ..., y_m

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

.

It follows that

L_f ≤ U_f.

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

x₀, ..., x_n

and

t₀,...,t_m-1

together make a tagged partition (as in the definition of the Riemann integral), and if the Riemann sum of f corresponding to x_n and If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...

t₀,...,t_m-1

is R, then

.

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