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Azagra Rueda, Daniel and Muñoz-Fernández, Gustavo A. and Seoane-Sepúlveda, Juan B. 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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Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X08012419

## Abstract

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 |
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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: | 28 Nov 2016 08:57 |

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