Riemann integrability versus weak continuity

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• 作者：Gonzalo Martí ; nez-Cervantes ^{gonzalo.martinez2@um.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Riemann integral ; Lebesgue property ; Weak Lebesgue property ; Banach space
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：438
• 期：2
• 页码：840-855
• 全文大小：439 K

文摘

In this paper we focus on the relation between Riemann integrability and weak continuity. A Banach space X   is said to have the weak Lebesgue property if every Riemann integrable function from an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000883&_mathId=si1.gif&_user=111111111&_pii=S0022247X16000883&_rdoc=1&_issn=0022247X&md5=d4905ac9b9f95d39a7e607acd31a0308" title="Click to view the MathML source">[0,1]an>an class="mathContainer hidden">an class="mathCode">$[0,1]$an>an>an> into X   is weakly continuous almost everywhere. We prove that the weak Lebesgue property is stable under an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16000883&_mathId=si2.gif&_user=111111111&_pii=S0022247X16000883&_rdoc=1&_issn=0022247X&md5=f7be662a2c7a83e3f911dd213cc94486" title="Click to view the MathML source">ℓ₁an>an class="mathContainer hidden">an class="mathCode">${\ell }_1$an>an>an>-sums and obtain new examples of Banach spaces with and without this property. Furthermore, we characterize Dunford–Pettis operators in terms of Riemann integrability and provide a quantitative result about the size of the set of τ-continuous nonRiemann integrable functions, with τ a locally convex topology weaker than the norm topology.