that function and the X axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions...

aire ("Integral, length, area") at the University of Nancy during 1902. Henri Lebesgue was born on 28 June 1875 in Beauvais, Oise. Lebesgue's father was...

early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; it is more general than...

theorem is named for Henri Lebesgue. For a Lebesgue integrable real or complex-valued function f on Rn, the indefinite integral is a set function which maps...

Just like the Lebesgue version of (classical) integrals, one can compute product integrals by approximating them with the product integrals of simple functions...

Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued...

integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician Meier Hirsch [de] (also spelled Meyer Hirsch) in 1810...

monotonically grow. It is also called the Cantor ternary function, the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase...

rapid decreasing, that the integral Eq.1 converges for all real ξ {\displaystyle \xi } , and (using the Riemann–Lebesgue lemma) that the transformed...

(considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). This result was first proven by Mathias Lerch...

its introduction of the Lebesgue integral. It was in 1912, "the only textbook on analysis containing both Lebesgue integral and its application to Fourier...

are called the Lebesgue points of f. Using this theorem it follows that if f is an integrable function on the circle, the Poisson integral Trf tends pointwise...

Lebesgue integration, one can then define an integral for all Borel measurable functions f {\displaystyle f} on G {\displaystyle G} . This integral is...

function FX of X, with the expected value of g(X) now given by the Lebesgue–Stieltjes integral E ⁡ [ g ( X ) ] = ∫ − ∞ ∞ g ( x ) d F X ( x ) . {\displaystyle...

{\displaystyle |f(t)|\leq Ae^{B|t|}} ), the integral can be understood to be a (proper) Lebesgue integral. However, for many applications it is necessary...

bestemte Integraler (Solution de quelques problèmes à l'aide d'intégrales définies, Solution of a couple of problems by means of definite integrals)" (PDF)...

immediately from those of Lebesgue integral. Note that the letters "a.s." stand for "almost surely"—a central property of the Lebesgue integral. Basically, one...

d W t {\displaystyle \int f\;dW_{t}} be the Wiener integral, d t {\displaystyle dt} be the Lebesgue measure. Further let H {\displaystyle \mathbf {H} }...

greatly extended. Henri Lebesgue invented measure theory, based on earlier developments by Émile Borel, and used it to define integrals of all but the most...

In mathematics, the Riemann–Liouville integral associates with a real function f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } another...

theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability...

0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory, if f and g are functions such...

Stratonovich integral can readily be expressed in terms of the Itô integral, and vice versa. The main benefit of the Stratonovich integral is that it obeys...

development was the Lebesgue integral, an alternative to the Riemann integral introduced by Henri Lebesgue in 1904. The Lebesgue integral made it possible...

It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The integral is evaluated for all...

an integral on the real line that was very similar in construction and simplicity to the Riemann integral but which included the Lebesgue integral and...

the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence...

and in the path integral approach to the quantum mechanics of particles and fields. In an ordinary integral (in the sense of Lebesgue integration) there...

In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse f − 1 {\displaystyle...

Fundamental Theorem of Multivariate Calculus. Stokes' theorem says that the integral of a differential form ω {\displaystyle \omega } over the boundary ∂ Ω...