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| In [[measure theory|measure-theoretic]] [[Mathematical analysis|analysis]] and related branches of [[mathematics]], '''Lebesgue–Stieltjes integration''' generalizes [[Riemann–Stieltjes integral|Riemann–Stieltjes]] and [[Lebesgue integration]], preserving the many advantages of the former in a more general measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of [[bounded variation]] on the real line. The Lebesgue–Stieltjes measure is a [[regular Borel measure]], and conversely every regular Borel measure on the real line is of this kind.
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| Lebesgue–Stieltjes [[integral]]s, named for [[Henri Leon Lebesgue]] and [[Thomas Joannes Stieltjes]], are also known as '''Lebesgue–Radon integrals''' or just '''Radon integrals''', after [[Johann Radon]], to whom much of the theory is due. They find common application in [[probability theory|probability]] and [[stochastic process]]es, and in certain branches of [[mathematical analysis|analysis]] including [[potential theory]].
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| ==Definition==
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| The Lebesgue–Stieltjes integral
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| :<math>\int_a^b f(x)\,dg(x)</math> | |
| is defined when ''ƒ'' : [''a'',''b''] → '''R''' is [[Borel measure|Borel]]-[[measurable function|measurable]]
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| and [[bounded function|bounded]] and ''g'' : [''a'',''b''] → '''R''' is of [[bounded variation]] in [''a'',''b''] and right-continuous, or when ''ƒ'' is non-negative and ''g'' is [[monotone function|monotone]] and right-continuous. To start, assume that ''ƒ'' is non-negative and ''g'' is monotone non-decreasing and right-continuous. Define ''w''((''s'',''t'']) := ''g''(''t'') − ''g''(''s'') and ''w''({''a''}) := 0 (Alternatively, the construction works for ''g'' left-continuous, ''w''([''s'',''t'')) := ''g''(''t'') − ''g''(''s'') and ''w''({''b''}) := 0).
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| By [[Carathéodory's extension theorem]], there is a unique Borel measure μ<sub>''g''</sub> on
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| [''a'',''b''] which agrees with ''w'' on every interval ''I''. The measure μ<sub>''g''</sub> arises from an [[outer measure]] (in fact, a [[metric outer measure]]) given by
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| :<math>\mu_g(E) = \inf\left\{\sum_i \mu_g(I_i) \right\vert \left. E\subset \bigcup_i I_i \right\}</math>
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| the [[infimum]] taken over all coverings of ''E'' by countably many semiopen intervals. This measure is sometimes called<ref>Halmos (1974), Sec. 15</ref> the '''Lebesgue–Stieltjes measure''' associated with ''g''. | |
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| The Lebesgue–Stieltjes integral
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| :<math>\int_a^b f(x)\,dg(x)</math>
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| is defined as the [[Lebesgue integral]] of ''ƒ''
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| with respect to the measure μ<sub>''g''</sub> in the usual way. If ''g'' is non-increasing, then define
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| :<math>\int_a^b f(x)\,dg(x) := -\int_a^b f(x) \,d (-g)(x),</math>
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| the latter integral being defined by the preceding construction. | |
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| If ''g'' is of bounded variation and ''ƒ'' is bounded, then it is possible to write
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| :<math>dg(x)=dg_1(x)-dg_2(x)</math>
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| where {{nowrap|1=''g''<sub>1</sub>(''x'') := ''V''{{su|b=''a''|p=''x''}}''g''}} is the [[total variation]]
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| of ''g'' in the interval [''a'',''x''], and ''g''<sub>2</sub>(''x'') = ''g''<sub>1</sub>(''x'') − ''g''(''x''). | |
| Both ''g''<sub>1</sub> and ''g''<sub>2</sub> are monotone non-decreasing. Now the Lebesgue–Stieltjes integral with respect to ''g'' is defined by
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| :<math>\int_a^b f(x)\,dg(x) = \int_a^b f(x)\,dg_1(x)-\int_a^b f(x)\,dg_2(x),</math>
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| where the latter two integrals
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| are well-defined by the preceding construction.
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| ===Daniell integral===
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| An alternative approach {{harv|Hewitt|Stromberg|1965}} is to define the Lebesgue–Stieltjes integral as the [[Daniell integral]] that extends the usual Riemann–Stieltjes integral. Let ''g'' be a non-increasing right-continuous function on [''a'',''b''], and define ''I''(''ƒ'') to be the Riemann–Stieltjes integral
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| :<math>I(f) = \int_a^b f(x)\,dg(x)</math>
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| for all continuous functions ''ƒ''. The [[functional]] ''I'' defines a [[Radon measure]] on [''a'',''b'']. This functional can then be extended to the class of all non-negative functions by setting
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| :<math>\overline{I}(h) = \sup \{I(f) | f\in C[a,b], 0\le f\le h\}</math>
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| and
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| :<math>\overline{\overline{I}}(h) = \inf\{I(f) | f\in C[a,b], h\le f\}.</math>
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| For Borel measurable functions, one has
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| :<math>\overline{I}(h) = \overline{\overline{I}}(h),</math>
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| and either side of the identity then defines the Lebesgue–Stieltjes integral of ''h''. The outer measure μ<sub>''g''</sub> is defined via
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| :<math>\mu_g(A) = \overline{\overline{I}}(\chi_A)</math> | |
| where χ<sub>''A''</sub> is the [[indicator function]] of ''A''.
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| Integrators of bounded variation are handled as above by decomposing into positive and negative variations.
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| ==Example== | |
| Suppose that <math>\gamma:[a,b]\to\R^2</math> is a [[rectifiable curve]] in the plane
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| and <math>\rho:\R^2\to[0,\infty)</math> is Borel measurable. Then we may define the length
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| of <math>\gamma</math> with respect to the Euclidean metric weighted by <math>\rho</math> to
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| be <math>\int_a^b \rho(\gamma(t))\,d\ell(t)</math>, where <math>\ell(t)</math> is the length | |
| of the restriction of <math>\gamma</math> to <math>[a,t]</math>.
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| This is sometimes called the <math>\rho</math>-length of <math>\gamma</math>.
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| This notion is quite useful for
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| various applications: for example, in muddy terrain the speed in which a person can move may
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| depend on how deep the mud is. If <math>\rho(z)</math> denotes the inverse of the walking speed
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| at or near <math>z</math>, then the <math>\rho</math>-length of <math>\gamma</math> is the
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| time it would take to traverse <math>\gamma</math>. The concept of [[extremal length]] uses
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| this notion of the <math>\rho</math>-length of curves and is useful in the study of
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| [[conformal map]]pings.
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| ==Integration by parts==
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| A function <math>f</math> is said to be "regular" at a point <math>a</math> if the right and left hand limits <math>f(a+)</math> and <math>f(a-)</math> exist, and the function takes the average value,
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| :<math>f(a)=\frac{1}{2}\left(f(a-)+f(a+)\right),</math>
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| at the limiting point. Given two functions <math>U</math> and <math>V</math> of finite variation, if at each point either <math>U</math> or <math>V</math> is continuous, or if both <math>U</math> and <math>V</math> are regular, then there is an [[integration by parts]] formula for the Lebesgue–Stieltjes integral:
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| :<math>\int_a^b U\,dV+\int_a^b V\,dU=U(b+)V(b+)-U(a-)V(a-),</math>
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| where <math>b>a</math>. Under a slight generalization of this formula, the extra conditions on <math>U</math> and <math>V</math> can be dropped.<ref>{{cite journal |last=Hewitt |first=Edwin |year=1960 |month=5 |title=Integration by Parts for Stieltjes Integrals |journal=The American Mathematical Monthly |volume=67 |issue=5 |pages=419–423 |jstor=2309287 |doi=10.2307/2309287 }}</ref> | |
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| An alternative result, of significant importance in the theory of [[Stochastic calculus]] is the following. Given two functions <math>U</math> and <math>V</math> of finite variation, which are both right-continuous and have left-limits (they are [[cadlag]] functions) then
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| :<math>U(t)V(t) = U(0)V(0) + \int_{(0,t]} U(s-)\,dV(s)+\int_{(0,t]} V(s-)\,dU(s)+\sum_{u\in (0,t]} \Delta U_u \Delta V_u,</math>
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| where <math>\Delta U_t= U(t)-U(t-)</math>. This result can be seen as a precursor to [[Itō's lemma]], and is of use in the general theory of Stochastic integration. The final term is <math>\Delta U(t)\Delta V(t)= d[U,V]</math>, which arises from the quadratic covariation of <math>U</math> and <math> V </math>. (The earlier result can then be seen as a result pertaining to the [[Stratonovich integral]].)
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| ==Related concepts==
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| ===Lebesgue integration===
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| When ''g''(''x'') = ''x'' for all real ''x'', then μ<sub>''g''</sub> is the [[Lebesgue measure]], and the Lebesgue–Stieltjes integral of ''f'' with respect to ''g'' is equivalent to the [[Lebesgue integral]] of ''f''.
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| ===Riemann–Stieltjes integration and probability theory===
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| Where ''f'' is a [[continuous function|continuous]] real-valued function of a real variable and ''v'' is a non-decreasing real function, the Lebesgue–Stieltjes integral is equivalent to the [[Riemann–Stieltjes integral]], in which case we often write
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| :<math>\int_a^b f(x) \, dv(x)</math>
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| for the Lebesgue–Stieltjes integral, letting the measure μ<sub>''v''</sub> remain implicit. This is particularly common in [[probability theory]] when ''v'' is the [[cumulative distribution function]] of a real-valued random variable ''X'', in which case
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| :<math>\int_{-\infty}^\infty f(x) \, dv(x) = \mathrm{E}[f(X)].</math>
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| (See the article on [[Riemann–Stieltjes integral|Riemann–Stieltjes integration]] for more detail on dealing with such cases.)
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Citation | last1=Halmos | first1=Paul R. | author1-link=Paul R. Halmos | title=Measure Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90088-9 | year=1974}}
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| * {{citation|title=Real and abstract analysis|publisher=Springer-Verlag|first1=Edwin|last1=Hewitt|first2=Karl|last2=Stromberg|year=1965}}.
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| * Saks, Stanislaw (1937) ''[http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10 Theory of the Integral.]''
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| * Shilov, G. E., and Gurevich, B. L., 1978. ''Integral, Measure, and Derivative: A Unified Approach'', Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8.
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| {{integral}}
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| {{DEFAULTSORT:Lebesgue-Stieltjes integration}}
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| [[Category:Definitions of mathematical integration]]
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| [[de:Stieltjesintegral]]
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| [[id:Integrasi Lebesgue-Stieltjes]]
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| [[it:Integrale di Lebesgue-Stieltjes]]
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| [[es:Integración de Lebesgue–Stieltjes]]
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