Editing Integral (section)

==Properties==

===Linearity===
The collection of Riemann-integrable functions on a closed interval {{math|[''a'', ''b'']}} forms a [[vector space]] under the operations of [[pointwise addition]] and multiplication by a scalar, and the operation of integration

: <math> f \mapsto \int_a^b f(x) \; dx</math>

is a [[linear functional]] on this vector space. Thus, the collection of integrable functions is closed under taking [[linear combination]]s, and the integral of a linear combination is the linear combination of the integrals:<ref name=":0">{{Harvnb|Apostol|1967|p=80}}.</ref>

: <math> \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,</math>

Similarly, the set of [[Real number|real]]-valued Lebesgue-integrable functions on a given [[Measure (mathematics)|measure space]] {{mvar|E}} with measure {{mvar|μ}} is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral

: <math> f\mapsto \int_E f \, d\mu </math>

is a linear functional on this vector space, so that:<ref name=":3" />

: <math> \int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu. </math>

More generally, consider the vector space of all [[measurable function]]s on a measure space {{math|(''E'',''μ'')}}, taking values in a [[Locally compact space|locally compact]] [[Complete metric space|complete]] [[topological vector space]] {{mvar|V}} over a locally compact [[Topological ring|topological field]] {{math|''K'', ''f'' : ''E'' → ''V''}}. Then one may define an abstract integration map assigning to each function {{mvar|f}} an element of {{mvar|V}} or the symbol {{math|''∞''}},

: <math> f\mapsto\int_E f \,d\mu, \,</math>

that is compatible with linear combinations.<ref>{{Harvnb|Rudin|1987|p=54}}.</ref> In this situation, the linearity holds for the subspace of functions whose integral is an element of {{mvar|V}} (i.e. "finite"). The most important special cases arise when {{mvar|K}} is {{math|'''R'''}}, {{math|'''C'''}}, or a finite extension of the field {{math|'''Q'''<sub>''p''</sub>}} of [[p-adic number]]s, and {{mvar|V}} is a finite-dimensional vector space over {{mvar|K}}, and when {{math|''K'' {{=}} '''C'''}} and {{mvar|V}} is a complex [[Hilbert space]].

Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of [[Daniell integral|Daniell]] for the case of real-valued functions on a set {{mvar|X}}, generalized by [[Nicolas Bourbaki]] to functions with values in a locally compact topological vector space. See {{Harvnb|Hildebrandt|1953}} for an axiomatic characterization of the integral.

=== Inequalities ===
A number of general inequalities hold for Riemann-integrable [[Function (mathematics)|functions]] defined on a [[Closed set|closed]] and [[Bounded set|bounded]] [[Interval (mathematics)|interval]] {{closed-closed|''a'', ''b''}} and can be generalized to other notions of integral (Lebesgue and Daniell).

* ''Upper and lower bounds.'' An integrable function {{mvar|f}} on {{closed-closed|''a'', ''b''}}, is necessarily [[Bounded function|bounded]] on that interval. Thus there are [[real number]]s {{mvar|m}} and {{mvar|M}} so that {{math|''m'' ≤ ''f'' (''x'') ≤ ''M''}} for all {{mvar|x}} in {{closed-closed|''a'', ''b''}}. Since the lower and upper sums of {{mvar|f}} over {{closed-closed|''a'', ''b''}} are therefore bounded by, respectively, {{math|''m''(''b'' − ''a'')}} and {{math|''M''(''b'' − ''a'')}}, it follows that <math display="block"> m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a). </math>
* ''Inequalities between functions.''<ref>{{Harvnb|Apostol|1967|p=81}}.</ref> If {{math|''f''(''x'') ≤ ''g''(''x'')}} for each {{mvar|x}} in {{closed-closed|''a'', ''b''}} then each of the upper and lower sums of {{mvar|f}} is bounded above by the upper and lower sums, respectively, of {{mvar|g}}. Thus <math display="block"> \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx. </math> This is a generalization of the above inequalities, as {{math|''M''(''b'' − ''a'')}} is the integral of the constant function with value {{mvar|M}} over {{closed-closed|''a'', ''b''}}. In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if {{math|''f''(''x'') < ''g''(''x'')}} for each {{mvar|x}} in {{closed-closed|''a'', ''b''}}, then <math display="block"> \int_a^b f(x) \, dx < \int_a^b g(x) \, dx. </math>
* ''Subintervals.'' If {{closed-closed|''c'', ''d''}} is a subinterval of {{closed-closed|''a'', ''b''}} and {{math|''f''&hairsp;(''x'')}} is non-negative for all {{mvar|x}}, then <math display="block"> \int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx. </math>
* ''Products and absolute values of functions.'' If {{mvar|f}} and {{mvar|g}} are two functions, then we may consider their [[pointwise product]]s and powers, and [[absolute value]]s: <math display="block">
 (fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; |f| (x) = |f(x)|.</math> If {{mvar|f}} is Riemann-integrable on {{closed-closed|''a'', ''b''}} then the same is true for {{math|{{abs|''f''}}}}, and <math display="block">\left| \int_a^b f(x) \, dx \right| \leq \int_a^b | f(x) | \, dx. </math> Moreover, if {{mvar|f}} and {{mvar|g}} are both Riemann-integrable then {{math|''fg''}} is also Riemann-integrable, and <math display="block">\left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right). </math> This inequality, known as the [[Cauchy–Schwarz inequality]], plays a prominent role in [[Hilbert space]] theory, where the left hand side is interpreted as the [[Inner product space|inner product]] of two [[Square-integrable function|square-integrable]] functions {{mvar|f}} and {{mvar|g}} on the interval {{closed-closed|''a'', ''b''}}.
* ''Hölder's inequality''.<ref name=":4">{{Harvnb|Rudin|1987|p=63}}.</ref> Suppose that {{mvar|p}} and {{mvar|q}} are two real numbers, {{math|1 ≤ ''p'', ''q'' ≤ ∞}} with {{math|1={{sfrac|1|''p''}} + {{sfrac|1|''q''}} = 1}}, and {{mvar|f}} and {{mvar|g}} are two Riemann-integrable functions. Then the functions {{math|{{abs|''f''}}<sup>''p''</sup>}} and {{math|{{abs|''g''}}<sup>''q''</sup>}} are also integrable and the following [[Hölder's inequality]] holds: <math display="block">\left|\int f(x)g(x)\,dx\right| \leq
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.</math> For {{math|1=''p'' = ''q'' = 2}}, Hölder's inequality becomes the Cauchy–Schwarz inequality.
* ''Minkowski inequality''.<ref name=":4" /> Suppose that {{math|''p'' ≥ 1}} is a real number and {{mvar|f}} and {{mvar|g}} are Riemann-integrable functions. Then {{math|{{abs| ''f'' }}<sup>''p''</sup>, {{abs| ''g'' }}<sup>''p''</sup>}} and {{math|{{abs| ''f'' + ''g'' }}<sup>''p''</sup>}} are also Riemann-integrable and the following [[Minkowski inequality]] holds: <math display="block">\left(\int \left|f(x)+g(x)\right|^p\,dx \right)^{1/p} \leq
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} +
\left(\int \left|g(x)\right|^p\,dx \right)^{1/p}.</math> An analogue of this inequality for Lebesgue integral is used in construction of [[Lp space|L<sup>p</sup> spaces]].

=== Conventions ===
In this section, {{mvar|f}} is a [[Real number|real-valued]] Riemann-integrable [[Function (mathematics)|function]]. The integral

: <math> \int_a^b f(x) \, dx </math>

over an interval {{math|[''a'', ''b'']}} is defined if {{math|''a'' &lt; ''b''}}. This means that the upper and lower sums of the function {{mvar|f}} are evaluated on a partition {{math|''a'' {{=}} ''x''<sub>0</sub> ≤ ''x''<sub>1</sub> ≤ . . . ≤ ''x''<sub>''n''</sub> {{=}} ''b''}} whose values {{math|''x''<sub>''i''</sub>}} are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating {{mvar|f}} within intervals {{math|[''x''<sub> ''i''</sub> , ''x''<sub> ''i'' +1</sub>]}} where an interval with a higher index lies to the right of one with a lower index. The values {{mvar|a}} and {{mvar|b}}, the end-points of the [[Interval (mathematics)|interval]], are called the [[limits of integration]] of {{mvar|f}}. Integrals can also be defined if {{math|''a'' &gt; ''b''}}:''<ref name=":1" />''

:<math>\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx. </math>

With {{math|''a'' {{=}} ''b''}}, this implies:

:<math>\int_a^a f(x) \, dx = 0. </math>

The first convention is necessary in consideration of taking integrals over subintervals of {{math|[''a'', ''b'']}}; the second says that an integral taken over a degenerate interval, or a [[Point (geometry)|point]], should be [[0 (number)|zero]]. One reason for the first convention is that the integrability of {{mvar|f}} on an interval {{math|[''a'', ''b'']}} implies that {{mvar|f}} is integrable on any subinterval {{math|[''c'', ''d'']}}, but in particular integrals have the property that if {{mvar|c}} is any [[Element (mathematics)|element]] of {{math|[''a'', ''b'']}}, then:''<ref name=":0" />''

:<math> \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.</math>

With the first convention, the resulting relation

: <math>\begin{align}
 \int_a^c f(x) \, dx &{}= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \\
 &{} = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx
\end{align}</math>

is then well-defined for any cyclic permutation of {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}.