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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 of the present topic is due. They find common application in Probability and Stochastic Process es, and in certain branches of Analysis including Potential Theory .


FORMAL CONSTRUCTION


In order to define the Lebesgue-Stieltjes integral, we will begin by associating a measure, μ''w'', with a Non-negative , Additive Function of an Interval , ''w''(''I''), which is of Bounded Variation . Let (Ω, ''F'') be a Measurable Space such that ''w'' has support on ''F'', then define
:(1) \quad \mu_w(E) := \inf \left\{\sum_j w(I_j) : E \subseteq \Omega, \, E \subset \bigcup_j I_j ight\},
(the lower bound over all sequences of intervals {''I''''j''}). Note that it is possible to show that μ''w'' is an Outer Measure .

We may now proceed to construct the Lebesgue-Stieltjes integral of a non-negative, Measurable Function in a similar fashion to the Construction of the corresponding Lebesgue integral. If (Ω, ''F'', μ''w'') is a Measure Space , then we can define the integral of any Simple Function ''s'' = Σ''i'' ''a''i1''A''''i'' (where 1''A'' is the Indicator Function of ''A'') as
:\int s \, d\mu_w = \sum_i a_i \mu_w(A_i).
Then, if ''f'' is a μ''w''-measurable map, ''f'':(Ω, ''F'') → +∞ , we can define the integral of ''f'' with respect to μ''w'' over ''E'' ⊆ Ω, as
:(2) \quad \int_E f \, d\mu_w = \sup\left\{\int s\,d\mu_w^E : s < f, s\ \mbox{simple}\, ight\},
where μ''w''''E''(·) = μ''w''(''E''∩·) on ''E'' and 0 otherwise. (If ''E'' = Ω, μ''w''Ω = μ''w''.)

It is often required, of course, to compute the integral of arbitrary measurable functions ''f'':(Ω, ''F'') → '''R'''∪{-∞, +∞}, but (as for the Lebesgue integral) we may construct these from two non-negative functions. If ''g'':(Ω, ''F'') → +∞ and ''h'':(Ω, ''F'') → +∞ such that ''g'' = max(0,''f'') and ''h'' = max(-''f'',0), then clearly ''f'' = ''g'' - ''h'' and
:\int_E f \, d\mu_w = \int_E g \, d\mu_w - \int_E h \, d\mu_w.

We now have a theory of Lebesgue-Stieltjes integrals of arbitrary functions ''f'', with respect to measures μ''w'' associated with non-negative additive functions of an interval, of bounded variation. We generally want to deal with measures associated with arbitrary additive functions, however, so suppose that ''v'' is an arbitrary (i.e. possibly not non-negative) additive function of an interval, again of bounded variation. Let ''w''1 and ''w''2 denote the Upper And Lower Variation s of ''v'', respectively. Then
:(3) \quad \mu_v(E) = \mu_{w_1}(E) - \mu_{-w_2}(E),
where the measures μ''w''1 and μ-''w''2 are defined as in equation (1), above.

We are finally equipped to define the Lebesgue-Stieltjes integral of an arbitrary function ''f'' with respect to the measure associated with an arbitrary additive function of an interval, ''v'', which is of bounded variation.

Let ''g'' = max(0, ''f'') and ''h'' = max(-''f'', 0), and let ''w''1 and ''w''2 be the upper and lower variations of ''v'', respectively. Then if μ''v'' is defined according to equations (1) and (3), the ''Lebesgue-Stieltjes integral'' of ''f'' with respect to μ''v'' is
:\int_E f \, d\mu_v = \left(\int_E g \, d\mu_{w_1} - \int_E h \, d\mu_{w_1} ight) - \left(\int_E g \, d\mu_{-w_2} - \int_E h \, d\mu_{-w_2} ight),
where each of the integrals on the right hand side of this equation are defined according to (2).


RELATED CONCEPTS



Lebesgue integration


When μ''v'' is the Lebesgue Measure , then the Lebesgue-Stieltjes integral of ''f'' is equivalent to the Lebesgue Integral of ''f''.


Riemann-Stieltjes integration and probability theory


Where ''f'' is a 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
:\int_a^b f(x) \, dv(x)
for the Lebesgue-Stieltjes integral, letting the measure μ''v'' remain implicit. This is particularly common in Probability Theory when ''v'' is the Distribution Function of a real-valued random variable, in which case
:\int_{-\infty}^{\infty} f(x) \, dv(x) = \mathrm{E} {Link without Title} .
(See the article on Riemann-Stieltjes Integration for more detail on dealing with such cases.)


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