The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Lebesgue Integrable Function

a function to which the concept of the integral introduced by W. Lebesgue may be applied. In other words, an integrable function is a function whose Lebesgue integral, taken over a given set, is finite. The function must be Lebesgue measurable. Lebesgue integrable functions are often referred to simply as integrable functions. A square integrable function is a measurable function whose square is an integrable function.

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References in periodicals archive

a) The fractional integral of order q > 0 of a Lebesgue integrable function f(.) : (0, [infinity]) [right arrow] R is defined by

On a fractional differential inclusion with nonlocal Riemann-Liouville type integral boundary conditions

Let X be a compact metric space and f be a R-valued Lebesgue integrable function on X.

The orbit of probability density functions by the deformed tent map

Let f : [a, b] [right arrow] R be of bounded variation and g: [a, b] [right arrow] R a Lebesgue integrable function such that there exists the constants m and M with

If we assume that for the Lebesgue integrable function g, [[integral].sub.a.sup.[dot]]g (s) ds satisfies the condition

A generalisation of Cerone's identity and applications

Let (X, ||dot||) be a Banach space over the real or complex number field K, [OMEGA] [member of] [R.sup.n] a measurable set and [rho] : [OMEGA] [right arrow] [0, [infinity]) a Lebesgue integrable function with [[integral].sub.[OMEGA]] [rho] (x) dx = 1.

x [member of] [OMEGA] and [alpha] : [OMEGA] [right arrow] K a Lebesgue integrable function with [rho][alpha]f, [rho]f Bochner integrable functions on [OMEGA], then we have the sharp inequalities

Some Gruss type inequalities for vector-valued functions in Banach spaces and applications

Two traditional Banach spaces E = C[0,1] and [L.sup.1](0,1) are involved in this article, where E = C[0,1] and [L.sup.1](0,1) represent the spaces of the continuous functions and Lebesgue integrable functions equipped with the norms [parallel]u[parallel] = [max.sub.0 [less than or equal to] t [less than or equal to] 1] [parallel]u(t)[parallel] and [[parallel]u[parallel].sub.1] = [[integral].sup.1.sub.0] [parallel]u(t)[parallel] dt, respectively.

Positive Solutions for Singular Semipositone Fractional Differential Equation Subject to Multipoint Boundary Conditions

For two Lebesgue integrable functions f, h : [a, b] [right arrow] R we consider Cebysev functional

Integral Error Representation of Hermite Interpolating Polynomial and Related Inequalities for Quadrature Formulae

Some results concerning the Korovkin type approximation theorem in the space [L.sub.p] [a, b] of the Lebesgue integrable functions on a compact interval may be found in [2].

Lebesgue integrable functions f on [a, b] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[L.sub.p]-approximation via Abel convergence

However, we learn later that the family of Riemann integrable functions-denoted by R(f), is only a subset of Lebesgue integrable functions-denoted by L(f) and the family of Henstock-Kurzweil integrable functions-denoted by HK(f )-is an extension for Lebesgue integrable functions. In other words, we have

Numerical integration on some special Henstock-Kurzweil integrals

Assume f is p-times differentiable with [f.sup.(P)] [member of] [L.sup.1] (U), the space of Lebesgue integrable functions on U.

Quadrature over the sphere