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Riemann integrability and Lebesgue measurability of the composite function


Azagra Rueda, Daniel and Muñoz-Fernández, Gustavo A. and Seoane Sepúlveda, Juan Benigno and Sánchez de los Reyes, Víctor Manuel (2009) Riemann integrability and Lebesgue measurability of the composite function. Journal of Mathematical Analysisand applications, 354 . pp. 229-233. ISSN 0022-247X

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If f is continuous on the interval [a, b], g is Riemann integrable (resp. Lebesgue measurable) on the interval [alpha, beta] and g([alpha, beta]) subset of [a, b], then f o g is Riemann integrable (resp. measurable) on [alpha, beta]. A well-known fact, on the other hand, states that f o g might not be Riemann integrable (resp. measurable) when f is Riemann integrable (resp. measurable) and g is continuous. If c stands for the continuum, in this paper we construct a 2(c)-dimensional space V and a c-dimensional space W of, respectively, Riemann integrable functions and continuous functions such that, for every f is an element of V \ {0} and g is an element of W \ {0} . f o g is not Riemann integrable, showing that nice properties (such as continuity or Riemann integrability) can be lost, in a linear fashion, via the composite function. Similarly we construct a c-dimensional space W of continuous functions such that for every g is an element of W \ {0} there exists a c-dimensional space V of measurable functions such that f o g is not measurable for all f is an element of V \ {0}.

Item Type:Article
Uncontrolled Keywords:Lineability; Spaces; Algebrability; Sets; Spaceability; Riemann integrability; Lebesgue measurable function
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:14744
Deposited On:17 Apr 2012 10:26
Last Modified:14 Mar 2016 16:12

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