Editing Integral (section)

==Formal definitions==
{{multiple image
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| image = Integral Riemann sum.png
| alt1 = Riemann integral approximation example
| caption1 = Integral example with irregular partitions (largest marked in red)
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| caption2 = Riemann sums converging
}}There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but are also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals.

=== Riemann integral ===
{{Main|Riemann integral}}
The Riemann integral is defined in terms of [[Riemann sum]]s of functions with respect to ''tagged partitions'' of an interval.<ref>{{Harvnb|Anton|Bivens|Davis|2016|pp=286−287}}.</ref> A tagged partition of a [[closed interval]] {{math|[''a'', ''b'']}} on the real line 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'']}} into {{mvar|n}} sub-intervals {{math|[''x''<sub>''i''−1</sub>, ''x''<sub>''i''</sub>]}} indexed by {{mvar|i}}, each of which is "tagged" with a specific point {{math|''t''<sub>''i''</sub> ∈ [''x''<sub>''i''−1</sub>, ''x''<sub>''i''</sub>]}}. A ''Riemann sum'' of a function {{mvar|f}} with respect to such a tagged partition is defined as

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

thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, {{math|Δ<sub>''i''</sub> {{=}} ''x''<sub>''i''</sub>−''x''<sub>''i''−1</sub>}}. The ''mesh'' of such a tagged partition is the width of the largest sub-interval formed by the partition, {{math|max<sub>''i''{{=}}1...''n''</sub> Δ<sub>''i''</sub>}}. The ''Riemann integral'' of a function {{mvar|f}} over the interval {{math|[''a'', ''b'']}} is equal to {{mvar|S}} if:<ref>{{Harvnb|Krantz|1991|p=173}}.</ref>

: For all <math>\varepsilon > 0</math> there exists <math>\delta > 0</math> such that, for any tagged partition <math>[a, b]</math> with mesh less than <math>\delta</math>,

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

When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower) [[Darboux integral|Darboux sum]], suggesting the close connection between the Riemann integral and the [[Darboux integral]].

=== Lebesgue integral ===
{{Main|Lebesgue integration}}
[[File:Lebesgueintegralsimplefunctions finer-dotted.svg|alt=Comparison of Riemann and Lebesgue integrals|thumb|250x250px|Lebesgue integration]]
It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated.<ref>{{Harvnb|Rudin|1987|p=5}}.</ref>

Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus [[Henri Lebesgue]] introduced the integral bearing his name, explaining this integral thus in a letter to [[Paul Montel]]:<ref>{{Harvnb|Siegmund-Schultze|2008|p=796}}.</ref>

{{blockquote|I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.|title=|source=}}

As Folland puts it, "To compute the Riemann integral of {{mvar|f}}, one partitions the domain {{closed-closed|''a'', ''b''}} into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of {{mvar|f}} ".<ref>{{Harvnb|Folland|1999|pp=57–58}}.</ref> The definition of the Lebesgue integral thus begins with a [[Measure (mathematics)|measure]], μ. In the simplest case, the [[Lebesgue measure]] {{math|''μ''(''A'')}} of an interval {{math|1=''A'' = [''a'', ''b'']}} is its width, {{math|''b'' − ''a''}}, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist.<ref>{{Harvnb|Bourbaki|2004|p=IV.43}}.</ref> In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.

Using the "partitioning the range of {{mvar|f}} " philosophy, the integral of a non-negative function {{math|''f'' : '''R''' → '''R'''}} should be the sum over {{mvar|t}} of the areas between a thin horizontal strip between {{math|1=''y'' = ''t''}} and {{math|1=''y'' = ''t'' + ''dt''}}. This area is just {{math|''μ''{ ''x'' : ''f''(''x'') > ''t''} ''dt''}}. Let {{math|1=''f''<sup>∗</sup>(''t'') = ''μ''{ ''x'' : ''f''(''x'') > ''t'' }<nowiki/>}}. The Lebesgue integral of {{mvar|f}} is then defined by

: <math>\int f = \int_0^\infty f^*(t)\,dt</math>

where the integral on the right is an ordinary improper Riemann integral ({{math|''f''{{i sup|∗}}}} is a strictly decreasing positive function, and therefore has a [[well-defined]] improper Riemann integral).<ref>{{Harvnb|Lieb|Loss|2001|p=14}}.</ref> For a suitable class of functions (the [[measurable function]]s) this defines the Lebesgue integral.

A general measurable function {{mvar|f}} is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of {{mvar|f}} and the {{mvar|x}}-axis is finite:<ref>{{Harvnb|Folland|1999|p=53}}.</ref>

: <math>\int_E |f|\,d\mu < + \infty.</math>

In that case, the integral is, as in the Riemannian case, the difference between the area above the {{mvar|x}}-axis and the area below the {{mvar|x}}-axis:<ref name=":3">{{Harvnb|Rudin|1987|p=25}}.</ref>

: <math>\int_E f \,d\mu = \int_E f^+ \,d\mu - \int_E f^- \,d\mu</math>

where

: <math>\begin{alignat}{3}
 & f^+(x) &&{}={} \max \{f(x),0\} &&{}={} \begin{cases}
               f(x), & \text{if } f(x) > 0, \\
               0, & \text{otherwise,}
             \end{cases}\\
 & f^-(x) &&{}={} \max \{-f(x),0\} &&{}={} \begin{cases}
               -f(x), & \text{if } f(x) < 0, \\
               0, & \text{otherwise.}
             \end{cases}
\end{alignat}</math>

=== Other integrals ===
Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including:

* The [[Darboux integral]], which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the [[Riemann integral]]. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being easier to define than Riemann integrals.
* The [[Riemann–Stieltjes integral]], an extension of the Riemann integral which integrates with respect to a function as opposed to a variable.
* The [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes integral]], further developed by [[Johann Radon]], which generalizes both the Riemann–Stieltjes and Lebesgue integrals.
* The [[Daniell integral]], which subsumes the Lebesgue integral and [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes integral]] without depending on [[Measure (mathematics)|measures]].
* The [[Haar integral]], used for integration on locally compact topological groups, introduced by [[Alfréd Haar]] in 1933.
* The [[Henstock–Kurzweil integral]], variously defined by [[Arnaud Denjoy]], [[Oskar Perron]], and (most elegantly, as the gauge integral) [[Jaroslav Kurzweil]], and developed by [[Ralph Henstock]].
* The [[Khinchin integral]], named after [[Aleksandr Khinchin]].
* The [[Itô integral]] and [[Stratonovich integral]], which define integration with respect to [[semimartingale]]s such as [[Wiener process|Brownian motion]].
* The [[Young integral]], which is a kind of Riemann–Stieltjes integral with respect to certain functions of [[Bounded variation|unbounded variation]].
* The [[rough path]] integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both [[semimartingale]]s and processes such as the [[fractional Brownian motion]].
* The [[Choquet integral]], a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953.
* The [[Bochner integral]], a generalization of the Lebesgue integral to functions that take values in a [[Banach space]].