Because the integrand is bounded and is not Lebesgue integrable, it is not even Henstock – Kurzweil integrable.

Some of the technical deficiencies in Riemann integration can be remedied by the Riemann – Stieltjes integral, and most of these disappear with the Lebesgue integral.

On the other hand, by the Riemann – Lebesgue lemma, the weak limit exists and is zero.

* July 26 – Henri Lebesgue, French mathematician ( b. 1875 )

* June 28 – Henri Lebesgue, French mathematician ( d. 1941 )

Henri Léon Lebesgue ForMemRS (; June 28, 1875 – July 26, 1941 ) was a French mathematician most famous for his theory of integration, which was a generalization of the 17th century concept of integration — summing the area between an axis and the curve of a function defined for that axis.

representing a bounded function is a Fourier series, that the n < sup > th </ sup > Fourier coefficient tends to zero ( the Riemann – Lebesgue lemma ), and that a Fourier series is integrable term by term.

He also proves that the Riemann – Lebesgue lemma is a best possible result for continuous functions, and gives some treatment to Lebesgue constants.

In measure-theoretic analysis and related branches of mathematics, the Lebesgue – Stieltjes integral generalizes Riemann – Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework.

More general formulations of integration by parts exist for the Riemann – Stieltjes integral and Lebesgue – Stieltjes integral.

Fatou's lemma can be used to prove the Fatou – Lebesgue theorem and Lebesgue's dominated convergence theorem.

Then Lebesgue – Stieltjes integrals with respect to dH ( x ) are integrals with respect to the " reference measure " of the exponential family generated by H.

H ( x ) is a Lebesgue – Stieltjes integrator for the reference measure.

* For the Henstock – Kurzweil integral, improper integration is not necessary, and this is seen as a strength of the theory: it encompasses all Lebesgue integrable and improper Riemann integrable functions.

The Laplace – Stieltjes transform of a real-valued function g is given by a Lebesgue – Stieltjes integral of the form

In measure-theoretic analysis and related branches of mathematics, Lebesgue – Stieltjes integration generalizes 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.

Lebesgue – Stieltjes integrals, 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.

Moreover, if ( 1 ) is construed as a Lebesgue integral, then ( 1 ) is also undefined, because ( 1 ) is then defined simply as the difference ( 2 ) between positive and negative parts.

However, if ( 1 ) is construed as an improper integral rather than a Lebesgue integral, then ( 2 ) is undefined, and ( 1 ) is not necessarily well-defined.

Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral.

The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory.

In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space ; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral.

Measure theory provides the foundation for the modern notion of integral, the Lebesgue integral.

Formally, the Lebesgue integral provides the necessary analytic device.

The Lebesgue integral with respect to the measure δ satisfies

As a result, the latter notation is a convenient abuse of notation, and not a standard ( Riemann or Lebesgue ) integral.

Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations.

Lebesgue presents six conditions which it is desirable that the integral should satisfy, the last of which is " If the sequence f < sub > n </ sub >( x ) increases to the limit f ( x ), the integral of f < sub > n </ sub >( x ) tends to the integral of f ( x ).

" Lebesgue shows that his conditions lead to the theory of measure and measurable functions and the analytical and geometrical definitions of the integral.

Lebesgue integration has the property that every bounded function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree.

Thomas Joannes Stieltjes (, 29 December 1856 – 31 December 1894 ) was a Dutch mathematician.

The Thomas Stieltjes Institute for Mathematics at the University of Leiden is named after him, as is the Riemann – Stieltjes integral.

* Laplace – Stieltjes transform

* Riemann – Stieltjes integral

* Stieltjes – Wigert polynomials

* Chebyshev – Markov – Stieltjes inequalities

* Thomas Joannes Stieltjes, ( 1856 – 1894 ), mathematician

* Chebyshev – Markov – Stieltjes inequalities

However, similar ideas have been used before and go back at least to the introduction of the Riemann – Stieltjes integral which unifies sums and integrals.

# REDIRECT Riemann – Stieltjes integral

# REDIRECT Riemann – Stieltjes integral

In mathematics, the Riemann – Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.