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| In [[mathematics]], the '''regulated integral''' is a definition of [[Integral|integration]] for [[regulated function]]s, which are defined to be [[uniform norm|uniform limits]] of [[step function]]s. The use of the regulated integral instead of the [[Riemann integral]] has been advocated by [[Nicolas Bourbaki]] and [[Jean Dieudonné]].
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| ==Definition==
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| ===Definition on step functions===
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| Let [''a'', ''b''] be a fixed [[closed set|closed]], [[bounded set|bounded]] [[interval (mathematics)|interval]] in the [[real line]] '''R'''. A real-valued function ''φ'' : [''a'', ''b''] → '''R''' is called a '''step function''' if there exists a finite [[partition of an interval|partition]]
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| :<math>\Pi = \{ a = t_0 < t_1 < \cdots < t_k = b \}</math>
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| of [''a'', ''b''] such that ''φ'' is constant on each [[open set|open]] interval (''t''<sub>''i''</sub>, ''t''<sub>''i''+1</sub>) of Π; suppose that this constant value is ''c''<sub>''i''</sub> ∈ '''R'''. Then, define the '''integral''' of a step function ''φ'' to be
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| :<math>\int_a^b \varphi(t) \, \mathrm{d} t := \sum_{i = 0}^{k - 1} c_i | t_{i + 1} - t_i |.</math>
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| It can be shown that this definition is independent of the choice of partition, in that if Π<sub>1</sub> is another partition of [''a'', ''b''] such that ''φ'' is constant on the open intervals of Π<sub>1</sub>, then the numerical value of the integral of ''φ'' is the same for Π<sub>1</sub> as for Π.
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| ===Extension to regulated functions===
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| A function ''f'' : [''a'', ''b''] → '''R''' is called a '''[[regulated function]]''' if it is the uniform limit of a sequence of step functions on [''a'', ''b'']:
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| * there is a sequence of step functions (''φ''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub> such that || ''φ''<sub>''n''</sub> − ''f'' ||<sub>∞</sub> → 0 as ''n'' → ∞; or, equivalently,
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| * for all ''ε'' > 0, there exists a step function ''φ''<sub>''ε''</sub> such that || ''φ''<sub>''ε''</sub> − ''f'' ||<sub>∞</sub> < ''ε''; or, equivalently,
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| * ''f'' lies in the closure of the space of step functions, where the closure is taken in the space of all [[bounded function]]s [''a'', ''b''] → '''R''' and with respect to the [[supremum norm]] || - ||<sub>∞</sub>; or equivalently,
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| * for every ''t'' ∈ [''a'', ''b''), the right-sided limit
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| ::<math>f(t+) = \lim_{s \downarrow t} f(s)</math>
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| :exists, and, for every ''t'' ∈ (''a'', ''b''], the left-sided limit
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| ::<math>f(t-) = \lim_{s \uparrow t} f(s)</math>
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| :exists as well.
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| Define the '''integral''' of a regulated function ''f'' to be
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| :<math>\int_{a}^{b} f(t) \, \mathrm{d} t := \lim_{n \to \infty} \int_{a}^{b} \varphi_{n} (t) \, \mathrm{d} t,</math>
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| where (''φ''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub> is any sequence of step functions that converges uniformly to ''f''.
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| One must check that this limit exists and is independent of the chosen sequence, but this
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| is an immediate consequence of the [[continuous linear extension]] theorem of elementary | |
| functional analysis: a [[bounded linear operator]] ''T''<sub>0</sub> defined on a [[dense (topology)|dense]] [[linear subspace]] ''E''<sub>0</sub> of a [[normed linear space]] ''E'' and taking values in a Banach space ''F'' extends uniquely to a bounded linear operator ''T'' : ''E'' → ''F'' with the same (finite) [[operator norm]].
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| ==Properties of the regulated integral==
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| * The integral is a [[linear operator]]: for any regulated functions ''f'' and ''g'' and constants ''α'' and ''β'',
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| ::<math>\int_{a}^{b} \alpha f(t) + \beta g(t) \, \mathrm{d} t = \alpha \int_{a}^{b} f(t) \, \mathrm{d} t + \beta \int_{a}^{b} g(t) \, \mathrm{d} t.</math>
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| * The integral is also a [[bounded operator]]: every regulated function ''f'' is bounded, and if ''m'' ≤ ''f''(''t'') ≤ ''M'' for all ''t'' ∈ [''a'', ''b''], then
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| ::<math>m | b - a | \leq \int_{a}^{b} f(t) \, \mathrm{d} t \leq M | b - a |.</math>
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| : In particular:
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| ::<math>\left| \int_{a}^{b} f(t) \, \mathrm{d} t \right| \leq \int_{a}^{b} | f(t) | \, \mathrm{d} t.</math>
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| * Since step functions are integrable and the integrability and the value of a Riemann integral are compatible with uniform limits, the regulated integral is a special case of the Riemann integral.
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| ==Extension to functions defined on the whole real line==
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| It is possible to extend the definitions of step function and regulated function and the associated integrals to functions defined on the whole [[real line]]. However, care must be taken with certain technical points:
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| * the partition on whose open intervals a step function is required to be constant is allowed to be a countable set, but must be a [[discrete set]], i.e. have no [[limit point]]s;
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| * the requirement of uniform convergence must be loosened to the requirement of uniform convergence on [[compact space|compact sets]], i.e. [[closed set|closed]] and [[bounded set|bounded]] intervals;
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| * not every [[bounded function]] is integrable (e.g. the function with constant value 1). This leads to a notion of [[Locally integrable function|local integrability]].
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| ==Extension to vector-valued functions==
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| The above definitions go through ''[[mutatis mutandis]]'' in the case of functions taking values in a [[normed vector space]] ''X''.
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| ==See also==
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| * [[Lebesgue integration|Lebesgue integral]]
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| * [[Riemann integral]]
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| ==References==
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| *{{cite journal | author=Berberian, S.K. | title=Regulated Functions: Bourbaki's Alternative to the Riemann Integral | journal=The American Mathematical Monthly | year=1979 | doi=10.2307/2321526 | volume=86 | pages=208 | jstor=2321526 | issue=3 | publisher=Mathematical Association of America }}
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| *{{cite book | last=Gordon | first=Russell A. | title=The integrals of Lebesgue, Denjoy, Perron, and Henstock | series=Graduate Studies in Mathematics, 4 | publisher=American Mathematical Society | location=Providence, RI | year=1994 | isbn=0-8218-3805-9 }}
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| {{integral}}
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| {{Functional Analysis}}
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| [[Category:Definitions of mathematical integration]]
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