graph of that function and the X axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, is one way to make this concept rigorous...

Riemann integral for the Lebesgue integral. The definition of the Lebesgue integral is not obviously a generalization of the Riemann integral, but it...

measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which...

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...

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...

a double integral by using an iterated integral. It was introduced by Guido Fubini in 1907. The theorem states that if a function is Lebesgue integrable...

inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The...

generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral. In particular, a function is Lebesgue integrable over a...

(A)} . Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were...

Improper integrals can also be evaluated in the context of complex numbers, in higher dimensions, and in other theoretical frameworks such as Lebesgue integration...

between (1) and (3) is known as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue. For an equivalent definition in terms of measures see...

definition of the Lebesgue integral for a non-negative function). Remark 4. The proof below does not use any properties of the Lebesgue integral except those...

In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions...

Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical...

analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space...

algebra) Borel algebra Borel measure Indicator function Lebesgue measure Lebesgue integration Lebesgue's density theorem Counting measure Complete measure Haar...

of the Lebesgue integral is that it requires the initial development of a workable measure theory before any useful results for the integral can be obtained...

mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of a multidimensional Lebesgue integral to functions that take values...

finite products. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which...

not an integral in the Lebesgue sense; the word "integral" is used because the Berezin integral has properties analogous to the Lebesgue integral and because...

discontinuities of Lebesgue measure 0, then an antiderivative may be found by integration in the sense of Lebesgue. In fact, using more powerful integrals like the...

}{2}}.} This integral is not absolutely convergent, meaning | sin ⁡ x x | {\displaystyle \left|{\frac {\sin x}{x}}\right|} has infinite Lebesgue or Riemann...

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...

mathematics, Lebesgue integrability may refer to: Whether the Lebesgue integral of a function is defined; this is what is most often meant. The Lebesgue integrability...

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...

stage in the development of theories of integration, such as the Lebesgue integral, because it is easy to define integration for a simple function and...

mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and...

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...

{\displaystyle f:\mathbb {R} \to \mathbb {C} } is called (Lebesgue) integrable if the Lebesgue integral of its absolute value is finite: ‖ f ‖ 1 = ∫ R | f (...

(informal) Frobenius integrability Riemann-integrability Lebesgue-integrability; see Lebesgue integral Liouville-integrability Darboux-integrability Integrable...