Warning: notations are only consistent within one paragraph!
A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is closed downwards (meaning that whenever a is in A and x ¡Ü a, then x is in A as well), B is closed upwards and A has no maximum.
The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers. A typical Dedekind cut of the rational numbers is given by A = { a in Q : a2 < 2 }, B = { b in Q : b2 ¡Ý 2 }. This cut represents the real number ¡Ì 2 in Dedekind's construction.
The existence of representation by Dedekind-cuts of a dense subset is equivalent to separability of a metric space.
A more archaic term for integration is quadrature. Note quadrature has three different meanings in mathematics. Integration by quadrature either means solving an integral analytically (i.e., symbolically in terms of known functions), or solving of an integral numerically (e.g., Gaussian quadrature, quadrature formulas). Ueberhuber (1997, p. 71) uses the word "quadrature" to mean numerical computation of a univariate integral, and "cubature" to mean numerical computation of a multiple integral. The word quadrature is also used to mean squaring: the construction of a square using only compass and straightedge which has the same area as a given geometric figure. If quadrature is possible for a plane figure, it is said to be quadrable.
The Riemann integral is defined on interval in R. It can certainly be defined for functions whose domain are "intervals" in n-dimensional Euclidean space. But a Riemann integral is dependent on partitions, which depend on the structure of the real line. Therefore, you can not define a Riemann integrable for functions defined on more abstract sets (even the set of natural numbers).
One of the limitations of the Riemann integral is that it is based on the concept of an "interval", or rather on the length of subintervals
[x_{j-1}, x_j]. We therefore need to find a generalization of the "length" concept of a set in the real line. That new "length" concept, which
we will call "measure", should satisfy two key conditions:
1.The new "measure" concept should be applicable to intervals, unions of intervals, and to more general sets (such as
a Cantor set). Ideally, it should be defined for all sets.
2.The new "measure" concept should share as many properties as possible with the standard length of an interval,
such as:
the 'measure' of a set should be non-negative
the 'measure' of an interval should be the length of that interval
the 'measure' of countably many disjoint sets should be the sum of the 'measures' of the individual sets
To define what will eventually be called Lebesgue Measure, we follow a two-stage strategy:
Stage One:
We will define a concept extending length that is defined for all sets (to satisfy condition 1 above)
Stage Two:
We will modify that concept so that it looks as close as possible to the standard length concept (to satisfy condition 2 above)
The stage-one concept is called outer measure, defined as follows:
If A is any subset of R, define the (Lebesgue) outer measure of A as:
m*(A) = inf { l(A_n) }
where the infimum is taken over all countable collections of open intervals
A_n such that A \in union of A_n and l(An) is the standard length of the interval An.
Outer measure is defined for all sets in R and has some of the properties we wanted our concept of measure to have. It has subadditivity property but does not have additive property for union of disjoint sets.
If the limit exists it is called the Lebesgue integral and the function is called Lebesgue integrable.