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.

Definition

A partition of an interval b is a finite sequence

a = x₀ < x₁ < x₂ < ... < x_n = b.

Each x_i+1 is called a subinterval of the partition. A refinement of the partition

x₀, ..., x_n

is a partition

y₀, ..., y_m

such that for every i with

0≤i ≤ 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:a,b→R be a function, and let

x₀, ..., x_n

be a partition of b. Let:

M_i = \max_{x\in x_i,x_{i+1}} f(x)

m_i = \min_{x\in x_i,x_{i+1}} f(x)

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

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

The lower Darboux sum of f with respect to

x₀, ..., x_n

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

The upper Darboux integral of f is

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

The lower Darboux integral of f is

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

If U_f = L_f, 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

y₀, ..., y_m

is a refinement of

x₀, ..., x_n,

then

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

and

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

x₀, ..., x_n and

y₀, ..., y_m

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

L_{f, x_0,\ldots,x_n} \le U_{f, y_0,\ldots,y_m}

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

t₀,...,t_m-1

is R, then

L_{f, x_0,\ldots,x_n} \le R \le U_{f, x_0,\ldots,x_n}

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.

External link

Article on the Darboux integral by the Science Fair Project Encyclopedia

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