Darboux integral

The Darboux integral is a formulation of integration in real analysis defined using upper and lower sums over partitions of an interval. It provides an order-theoretic approach to integration and is equivalent to the Riemann integral.

Definition

Darboux sums

Let $f:[a,b]\to ℝ$ be a bounded function. Let $$P=\{x_0,x_1,\dots ,x_n\},a=x_0<x_1<\cdots <x_n=b$$ be a partition of the interval $[a,b]$.

For each subinterval $[x_{i-1},x_i]$ of $P$, define:

• the infimum:

$$m_i=\underset{x\in [x_{i-1},x_i]}{\inf }f(x),$$

• the supremum:

$$M_i=\underset{x\in [x_{i-1},x_i]}{\sup }f(x).$$

The lower Darboux sum of $f$ with respect to $P$ is $$L(f,P)=\sum _{i=1}^n{m}_i(x_i-x_{i-1}),$$

and the upper Darboux sum is $$U(f,P)=\sum _{i=1}^n{M}_i(x_i-x_{i-1}).$$

The Darboux integral

Let $𝒫([a,b])$ denote the set of all partitions of $[a,b]$.

The lower Darboux integral of $f$ on $[a,b]$ is defined by $$\underset{\_}{\underset{a}{\overset{b}{\int }}}f=\underset{P\in 𝒫([a,b])}{\sup }L(f,P).$$

and the upper Darboux integral is defined by $$\overline{\underset{a}{\overset{b}{\int }}}f=\underset{P\in 𝒫([a,b])}{\inf }U(f,P).$$

If the upper and lower Darboux integrals are equal, then the Darboux integral of $f$ on $[a,b]$ is defined by their common value, that is, $$\underset{a}{\overset{b}{\int }}f=\underset{\_}{\underset{a}{\overset{b}{\int }}}f=\overline{\underset{a}{\overset{b}{\int }}}f.$$ In this case, function $f$ is said to be Darboux-integrable.