Aumann Integral of Multifunctions Valued in Quasy-Banach Spaces and Some of its Properties

Enkeleda Zajmi Kotonaj

doi:https://doi.org/10.14445/22315373/IJMTT-V12P510

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 12 | Number 1 | Year 2014 | Article Id. IJMTT-V12P510 | DOI : https://doi.org/10.14445/22315373/IJMTT-V12P510

Aumann Integral of Multifunctions Valued in Quasy-Banach Spaces and Some of its Properties

Enkeleda Zajmi Kotonaj

Citation :

Enkeleda Zajmi Kotonaj, "Aumann Integral of Multifunctions Valued in Quasy-Banach Spaces and Some of its Properties," International Journal of Mathematics Trends and Technology (IJMTT), vol. 12, no. 1, pp. 63-68, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V12P510

Abstract

We will study the integral of Aumann when the values of multifunctions are subsets of a quasy-Banach separable space. Initially we will see the extension of the Aumann integral in the case of multifunctions with values in quasy-normed spaces and in some instances of its existence illustrating with examples. We will see the following relation between measurable and integrable of multifunctions according to Aumann and some various operations which do not bring us out of class by Aumann integrable multifunctions. Here we can mention that, if the union action of integrable multifunctions by Aumann is again integrable, for cutting can’t say the same thing. We see also that is true the property of linearity of the Aumann׳s integral. Finally we are shifting focus to the limit and note that there is a rather interesting statement that in the case when the space X which defines sequence of multifunctions F_n is finite measure allows us to say that: Not only the limit of a sequence of integrable multifunctions by Aumann is the integrable multifunction but it’s true the equality ∫▒〖Fdμ=lim┬(n→+∞)⁡∫▒〖F_n dμ〗〗. This assertion is also based on a similar theorem known Fatou theorem, which noted that, can be extended to our case.

Keywords

Aumann integral, Quasy-Banach space, Convergence theorem of multifunctions

References

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