On integration in branch spaces

Yadav, Vijay and Paul, Tanmoy (2016) On integration in branch spaces. Masters thesis, Indian Institute of Technology Hyderabad.

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Abstract

One of the major characterizations for a real valued function to be Riemann integrable is that the function is continuous except a negligible set (in the sense of Lebesgue)Lebesgue property. Similar conclusion can be drawn for the class of functions taking values in nite dimensional spaces. Unfortunately the result is not necessarily true for arbitrary vector valued functions. The main thrust of this part of our work is to explore the nonavail-ability of this property in the Banach spaces. This work also extends to de ne various notions of integrations in Banach spaces, the interplay between these integrals and the properties obtained by the Banach spaces under the assumption of convergence of these integrals. Basically integrability of a function taking values over a Banach space leads to a vector valued measure, which is sometime relevant to discuss the hidden properties of Banach spaces. In Chapter 3 we study on various notions of integrations for vector valued functions, several examples are given which lacks the Lebesgue property. Towards the end of this Chapter we have listed a family of Banach spaces which lacks the above property, though 1 has this property.

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