Editing Real analysis (section)

===Integration===
Integration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface.  The basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as the ''[[method of exhaustion]]''. Generally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed.  By considering approximations consisting of a larger and larger ("infinite") number of smaller and smaller ("infinitesimal") pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value.

The spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann (or Darboux) sums converge to a common value as thinner and thinner rectangular slices ("refinements") are considered.  Though the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind.  Compared to the Riemann integral, the more sophisticated Lebesgue integral allows area (or length, volume, etc.; termed a "measure" in general) to be defined and computed for much more complicated and irregular subsets of Euclidean space, although there still exist "non-measurable" subsets for which an area cannot be assigned.

====Riemann integration====
{{Main|Riemann integral}}
The Riemann integral is defined in terms of [[Riemann sum]]s of functions with respect to tagged partitions of an interval. Let <math>[a,b]</math> be a [[Interval (mathematics)|closed interval]] of the real line; then a '''''tagged partition''''' <math>\cal{P}</math> of <math>[a,b]</math> is a finite sequence

:<math> a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!</math>

This partitions the interval <math>[a,b]</math> into <math>n</math> sub-intervals <math>[x_{i-1},x_i]</math> indexed by <math>i=1,\ldots, n</math>, each of which is "tagged" with a distinguished point <math>t_i\in[x_{i-1},x_i]</math>. For a function <math>f</math> bounded on <math>[a,b]</math>, we define the '''''Riemann sum''''' of <math>f</math> with respect to  tagged partition <math>\cal{P}</math> as

:<math>\sum_{i=1}^{n} f(t_i) \Delta_i, </math>

where <math>\Delta_i=x_i-x_{i-1}</math> is the width of sub-interval <math>i</math>.  Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. The '''''mesh''''' of such a tagged partition is the width of the largest sub-interval formed by the partition, <math display="inline">\|\Delta_i\| = \max_{i=1,\ldots, n}\Delta_i</math>. We say that the '''''Riemann integral''''' of <math>f</math> on <math>[a,b]</math> is <math>S</math> if for any <math>\varepsilon>0</math> there exists <math>\delta>0</math> such that, for any tagged partition <math>\cal{P}</math> with mesh <math>\| \Delta_i \| < \delta</math>, we have

::<math>\left| S - \sum_{i=1}^{n} f(t_i)\Delta_i \right| < \varepsilon.</math>

This is sometimes denoted <math display="inline">\mathcal{R}\int_{a}^b f=S</math>.  When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) '''''Darboux sum'''''. A function is '''''Darboux integrable''''' if the upper and lower [[Darboux integral|Darboux sums]] can be made to be arbitrarily close to each other for a sufficiently small mesh.  Although this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal.  In fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former.

The [[fundamental theorem of calculus]] asserts that integration and differentiation are inverse operations in a certain sense.

====Lebesgue integration and measure====
{{Main|Lebesgue integral}}
'''Lebesgue integration''' is a mathematical construction that extends the integral to a larger class of functions; it also extends the [[domain of a function|domains]] on which these functions can be defined.  The concept of a [[Measure (mathematics)|'''''measure''''']], an abstraction of length, area, or volume, is central to Lebesgue integral [[probability theory]].