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Riemann integrability and Lebesgue measurability of the composite function


Azagra Rueda, Daniel and Muñoz Fernández, Gustavo Adolfo 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:06 Feb 2014 10:08

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