# Lebesgue Integrable Function

## 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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a) The fractional integral of order q > 0 of a Lebesgue integrable function f(.
Let X be a compact metric space and f be a R-valued Lebesgue integrable function on X.
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].
and [rho] : [OMEGA] [right arrow] [0, [infinity]) is a Lebesgue integrable function with [[integral].
n] a measurable set and [rho] : [OMEGA] [right arrow] [0, [infinity]) a Lebesgue integrable function with [[integral].
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
p] [a, b] of the Lebesgue integrable functions on a compact interval may be found in .
Lebesgue integrable functions f on [a, b] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
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.
1] (U), the space of Lebesgue integrable functions on U.

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