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| In [[mathematics]], a '''regulated function''' (or '''ruled function''') is a "well-behaved" [[function (mathematics)|function]] of a single [[real number|real]] variable. Regulated functions arise as a class of [[integration (mathematics)|integrable functions]], and have several equivalent characterisations. Regulated functions were introduced by Georg Aumann in 1954; the corresponding [[regulated integral]] was promoted by the [[Bourbaki]] group, including [[Jean Dieudonné]].
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| ==Definition==
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| Let ''X'' be a [[Banach space]] with norm || - ||<sub>''X''</sub>. A function ''f'' : [0, ''T''] → ''X'' is said to be a '''regulated function''' if one (and hence both) of the following two equivalent conditions holds true {{harv|Dieudonné|1969|loc=§7.6}}:
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| * for every ''t'' in the [[interval (mathematics)|interval]] [0, ''T''], both the [[limit of a function|left and right limits]] ''f''(''t''−) and ''f''(''t''+) exist in ''X'' (apart from, obviously, ''f''(0−) and ''f''(''T''+));
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| * there exists a [[sequence (mathematics)|sequence]] of [[step function]]s ''φ''<sub>''n''</sub> : [0, ''T''] → ''X'' [[uniform convergence|converging uniformly]] to ''f'' (i.e. with respect to the [[supremum norm]] || - ||<sub>∞</sub>).
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| It requires a little work to show that these two conditions are equivalent. However, it is relatively easy to see that the second condition may be re-stated in the following equivalent ways:
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| * for every ''δ'' > 0, there is some step function ''φ''<sub>''δ''</sub> : [0, ''T''] → ''X'' such that
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| ::<math>\| f - \varphi_{\delta} \|_{\infty} = \sup_{t \in [0, T]} \| f(t) - \varphi_{\delta} (t) \|_{X} < \delta;</math>
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| * ''f'' lies in the [[closed set|closure]] of the space Step([0, ''T'']; ''X'') of all step functions from [0, ''T''] into ''X'' (taking closure with respect to the supremum norm in the space B([0, ''T'']; ''X'') of all bounded functions from [0, ''T''] into ''X'').
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| ==Properties of regulated functions==
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| Let Reg([0, ''T'']; ''X'') denote the [[Set (mathematics)|set]] of all regulated functions ''f'' : [0, ''T''] → ''X''.
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| * Sums and scalar multiples of regulated functions are again regulated functions. In other words, Reg([0, ''T'']; ''X'') is a [[vector space]] over the same [[field (mathematics)|field]] '''K''' as the space ''X''; typically, '''K''' will be the [[real number|real]] or [[complex number]]s. If ''X'' is equipped with an operation of multiplication, then products of regulated functions are again regulated functions. In other words, if ''X'' is a '''K'''-[[Algebra over a field|algebra]], then so is Reg([0, ''T'']; ''X'').
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| * The supremum norm is a [[norm (mathematics)|norm]] on Reg([0, ''T'']; ''X''), and Reg([0, ''T'']; ''X'') is a [[topological vector space]] with respect to the topology induced by the supremum norm.
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| * As noted above, Reg([0, ''T'']; ''X'') is the closure in B([0, ''T'']; ''X'') of Step([0, ''T'']; ''X'') with respect to the supremum norm.
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| * If ''X'' is a [[Banach space]], then Reg([0, ''T'']; ''X'') is also a Banach space with respect to the supremum norm.
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| * Reg([0, ''T'']; '''R''') forms an infinite-dimensional real [[Banach algebra]]: finite linear combinations and products of regulated functions are again regulated functions.
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| * Since a [[continuous function]] defined on a [[compact space]] (such as [0, ''T'']) is automatically [[uniformly continuous function|uniformly continuous]], every continuous function ''f'' : [0, ''T''] → ''X'' is also regulated. In fact, with respect to the supremum norm, the space ''C''<sup>0</sup>([0, ''T'']; ''X'') of continuous functions is a [[closed set|closed]] [[linear subspace]] of Reg([0, ''T'']; ''X'').
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| * If ''X'' is a [[Banach space]], then the space BV([0, ''T'']; ''X'') of functions of [[bounded variation]] forms a [[dense set|dense]] linear subspace of Reg([0, ''T'']; ''X''):
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| ::<math>\mathrm{Reg}([0, T]; X) = \overline{\mathrm{BV} ([0, T]; X)} \mbox{ w.r.t. } \| \cdot \|_{\infty}.</math>
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| * If ''X'' is a Banach space, then a function ''f'' : [0, ''T''] → ''X'' is regulated [[if and only if]] it is of [[Bounded variation#Weighted BV functions|bounded ''φ''-variation]] for some ''φ'':
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| ::<math>\mathrm{Reg}([0, T]; X) = \bigcup_{\varphi} \mathrm{BV}_{\varphi} ([0, T]; X).</math>
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| * If ''X'' is a [[separable space|separable]] [[Hilbert space]], then Reg([0, ''T'']; ''X'') satisfies a compactness theorem known as the [[Fraňková-Helly selection theorem]].
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| * The set of [[Discontinuity (mathematics)|discontinuities]] of a regulated function is [[countable]]: to see this it is sufficient to note that given <math> \epsilon > 0 </math>, the set of points at which the right and left limits differ by more than <math> \epsilon</math> is finite. In particular, the discontinuity set has [[measure zero]], from which it follows that a regulated function has a well-defined [[Riemann integral]].
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| * The integral, as defined on step functions in the obvious way, extends naturally to Reg([0, ''T'']; ''X'') by defining the integral of a regulated function to be the limit of the integrals of any sequence of step functions converging uniformly to it. This extension is [[well-defined]] and satisfies all of the usual properties of an integral. In particular, the [[regulated integral]]
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| ** is a [[bounded linear function]] from Reg([0, ''T'']; ''X'') to ''X''; hence, in the case ''X'' = '''R''', the integral is an element of the [[continuous dual space|space that is dual]] to Reg([0, ''T'']; '''R''');
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| ** agrees with the [[Riemann integral]].
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| ==References==
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| * {{citation
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| | first = Georg
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| | last = Aumann
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| | title = Reelle Funktionen
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| | language = German
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| | series = Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd LXVIII
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| | publisher = Springer-Verlag
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| | location = Berlin
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| | year = 1954
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| | pages = viii+416
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| }} {{MathSciNet| id = 0061652}}
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| * {{citation
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| | first = Jean
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| | last = Dieudonné
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| | authorlink = Jean Dieudonné
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| | title = Foundations of Modern Analysis
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| | publisher = Academic Press
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| | year = 1969
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| | pages = xviii+387
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| }} {{MathSciNet | id = 0349288}}
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| * {{citation
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| | last = Fraňková
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| | first = Dana
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| | title = Regulated functions
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| | journal = Math. Bohem.
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| | volume = 116
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| | year = 1991
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| | pages = 20–59
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| | issn = 0862-7959
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| | issue = 1
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| }} {{MathSciNet | id = 1100424}}
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| * {{citation
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| | last = Gordon
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| | first = Russell A.
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| | title = The Integrals of Lebesgue, Denjoy, Perron, and Henstock
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| | series = Graduate Studies in Mathematics, 4
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| | publisher = American Mathematical Society
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| | location = Providence, RI
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| | year = 1994
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| | pages = xii+395
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| | isbn = 0-8218-3805-9
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| }} {{MathSciNet | id = 1288751}}
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| * {{citation
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| | last = Lang
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| | first = Serge
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| | authorlink = Serge Lang
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| | title = Differential Manifolds
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| | edition = Second
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| | publisher = Springer-Verlag
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| | location = New York
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| | year = 1985
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| | pages = ix+230
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| | isbn = 0-387-96113-5
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| }} {{MathSciNet | id = 772023}}
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| [[Category:Real analysis]]
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| [[Category:Types of functions]]
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