|
|
| Line 1: |
Line 1: |
| 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]].
| | Hi there, I am Alyson Pomerleau and I think it seems quite good when you say it. For years she's been residing in Kentucky but her husband wants them to move. Distributing production is where my main earnings comes from and it's something I truly appreciate. What me and my family members adore is bungee leaping but I've been using on new issues lately.<br><br>My blog [http://kpupf.com/xe/talk/735373 psychic readings] |
| | |
| ==Definition==
| |
| ===Definition on step functions===
| |
| 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]]
| |
| | |
| :<math>\Pi = \{ a = t_0 < t_1 < \cdots < t_k = b \}</math>
| |
| | |
| 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
| |
| | |
| :<math>\int_a^b \varphi(t) \, \mathrm{d} t := \sum_{i = 0}^{k - 1} c_i | t_{i + 1} - t_i |.</math>
| |
| | |
| 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.
| |
| | |
| ===Extension to regulated functions===
| |
| 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'']:
| |
| * 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,
| |
| * 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,
| |
| * ''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,
| |
| * for every ''t'' ∈ [''a'', ''b''), the right-sided limit
| |
| ::<math>f(t+) = \lim_{s \downarrow t} f(s)</math>
| |
| :exists, and, for every ''t'' ∈ (''a'', ''b''], the left-sided limit
| |
| ::<math>f(t-) = \lim_{s \uparrow t} f(s)</math>
| |
| :exists as well.
| |
| | |
| Define the '''integral''' of a regulated function ''f'' to be
| |
| | |
| :<math>\int_{a}^{b} f(t) \, \mathrm{d} t := \lim_{n \to \infty} \int_{a}^{b} \varphi_{n} (t) \, \mathrm{d} t,</math>
| |
| | |
| where (''φ''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub> is any sequence of step functions that converges uniformly to ''f''.
| |
| | |
| One must check that this limit exists and is independent of the chosen sequence, but this
| |
| 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]].
| |
| | |
| ==Properties of the regulated integral==
| |
| * The integral is a [[linear operator]]: for any regulated functions ''f'' and ''g'' and constants ''α'' and ''β'',
| |
| | |
| ::<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>
| |
| | |
| * 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
| |
| | |
| ::<math>m | b - a | \leq \int_{a}^{b} f(t) \, \mathrm{d} t \leq M | b - a |.</math>
| |
| | |
| : In particular:
| |
| | |
| ::<math>\left| \int_{a}^{b} f(t) \, \mathrm{d} t \right| \leq \int_{a}^{b} | f(t) | \, \mathrm{d} t.</math>
| |
| | |
| * 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.
| |
| | |
| ==Extension to functions defined on the whole real line==
| |
| 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:
| |
| * 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;
| |
| * 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;
| |
| * 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]].
| |
| | |
| ==Extension to vector-valued functions==
| |
| The above definitions go through ''[[mutatis mutandis]]'' in the case of functions taking values in a [[normed vector space]] ''X''.
| |
| | |
| ==See also==
| |
| * [[Lebesgue integration|Lebesgue integral]]
| |
| * [[Riemann integral]]
| |
| | |
| ==References==
| |
| *{{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 }}
| |
| *{{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 }}
| |
| | |
| {{integral}}
| |
| {{Functional Analysis}}
| |
| | |
| [[Category:Definitions of mathematical integration]]
| |
Hi there, I am Alyson Pomerleau and I think it seems quite good when you say it. For years she's been residing in Kentucky but her husband wants them to move. Distributing production is where my main earnings comes from and it's something I truly appreciate. What me and my family members adore is bungee leaping but I've been using on new issues lately.
My blog psychic readings