On Valdivia strong version of Nikodym boundedness property

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• 作者：J. Ka̧kol^a ; ^b ; ^{kakol@amu.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; M. Ló ; pez-Pellicer^c ; ^{mlopezpe@mat.upv.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Finitely additive scalar measure ; (LF)(LF)-space ; Nikodym and strong Nikodym property ; Increasing tree ; σ-algebra ; Vector measure
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：1-17
• 全文大小：488 K

文摘

Following Schachermayer, a subset class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=3582d885eda53545858453b9e73957c3" title="Click to view the MathML source">Bclass="mathContainer hidden">class="mathCode">$\mathcal{B}$ of an algebra class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si165.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=1abcd6d9224a9050be4c3b7e8009b7a7" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$\mathcal{A}$ of subsets of Ω is said to have the N-property   if a class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=3582d885eda53545858453b9e73957c3" title="Click to view the MathML source">Bclass="mathContainer hidden">class="mathCode">$\mathcal{B}$-pointwise bounded subset M   of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si166.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=cb7317e73d232c8f1896a14bb3380556" title="Click to view the MathML source">ba(A)class="mathContainer hidden">class="mathCode">$ba(\mathcal{A})$ is uniformly bounded on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si165.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=1abcd6d9224a9050be4c3b7e8009b7a7" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$\mathcal{A}$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si166.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=cb7317e73d232c8f1896a14bb3380556" title="Click to view the MathML source">ba(A)class="mathContainer hidden">class="mathCode">$ba(\mathcal{A})$ is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si165.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=1abcd6d9224a9050be4c3b7e8009b7a7" title="Click to view the MathML source">Aclass="mathContainer hidden">class="mathCode">$\mathcal{A}$. Moreover class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=3582d885eda53545858453b9e73957c3" title="Click to view the MathML source">Bclass="mathContainer hidden">class="mathCode">$\mathcal{B}$ is said to have the strong N-property   if for each increasing countable covering class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si126.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=de0c69c5eb1194df64ed44bffab98ede" title="Click to view the MathML source">(B_m)_mclass="mathContainer hidden">class="mathCode">${({\mathcal{B}}_m)}_m$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=3582d885eda53545858453b9e73957c3" title="Click to view the MathML source">Bclass="mathContainer hidden">class="mathCode">$\mathcal{B}$ there exists class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=4db93add3190391b57e1b073132617e6" title="Click to view the MathML source">B_nclass="mathContainer hidden">class="mathCode">${\mathcal{B}}_n$ which has the N-property. The classical Nikodym–Grothendieck's theorem says that each σ  -algebra class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si62.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=ac600c82bfbe751f9c9cb4f6294129d5" title="Click to view the MathML source">Sclass="mathContainer hidden">class="mathCode">$\mathcal{S}$ of subsets of Ω has the N-property. The Valdivia's theorem stating that each σ  -algebra class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si62.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=ac600c82bfbe751f9c9cb4f6294129d5" title="Click to view the MathML source">Sclass="mathContainer hidden">class="mathCode">$\mathcal{S}$ has the strong N  -property motivated the main measure-theoretic result of this paper: We show that if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si8.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=40c302d2cf033238bf8d9c6c8bad3f35" title="Click to view the MathML source">(B_m1)_m1class="mathContainer hidden">class="mathCode">${({\mathcal{B}}_{m_1})}_{m_1}$ is an increasing countable covering of a σ  -algebra class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si62.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=ac600c82bfbe751f9c9cb4f6294129d5" title="Click to view the MathML source">Sclass="mathContainer hidden">class="mathCode">$\mathcal{S}$ and if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si9.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=d792a59f98cd89ef429ae10eb94f61a5" title="Click to view the MathML source">(B_{m1,m2,…,mp,mp+1})_mp+1class="mathContainer hidden">class="mathCode">${({\mathcal{B}}_{m_1,m_2,\dots ,m_p,m_{p+1}})}_{m_{p+1}}$ is an increasing countable covering of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si10.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=9f572e21c7c02ee83f96710334cc78ec" title="Click to view the MathML source">B_m1,m2,…,mpclass="mathContainer hidden">class="mathCode">${\mathcal{B}}_{m_1,m_2,\dots ,m_p}$, for each class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si11.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=f0e324a1a28d5d9aad112a2eec5eab4d" title="Click to view the MathML source">p,m_i∈Nclass="mathContainer hidden">class="mathCode">$p,m_i\in \mathbb{N}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si12.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=7fe64eb8b29ac47d586e6444ea38e3ed" title="Click to view the MathML source">1⩽i⩽pclass="mathContainer hidden">class="mathCode">$1⩽i⩽p$, then there exists a sequence class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si13.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=7aad002d3b88319f92bc33eb5fa3643d" title="Click to view the MathML source">(n_i)_iclass="mathContainer hidden">class="mathCode">${(n_i)}_i$ such that each class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si14.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=f226f3ffdebdf872b2d4e58acee67a0f" title="Click to view the MathML source">B_n1,n2,…,nrclass="mathContainer hidden">class="mathCode">${\mathcal{B}}_{n_1,n_2,\dots ,n_r}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si15.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=500639b6ba394d8a907a9db54b607820" title="Click to view the MathML source">r∈Nclass="mathContainer hidden">class="mathCode">$r\in \mathbb{N}$, has the strong N  -property. In particular, for each increasing countable covering class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si126.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=de0c69c5eb1194df64ed44bffab98ede" title="Click to view the MathML source">(B_m)_mclass="mathContainer hidden">class="mathCode">${({\mathcal{B}}_m)}_m$ of a σ  -algebra class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si62.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=ac600c82bfbe751f9c9cb4f6294129d5" title="Click to view the MathML source">Sclass="mathContainer hidden">class="mathCode">$\mathcal{S}$ there exists class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304413&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304413&_rdoc=1&_issn=0022247X&md5=4db93add3190391b57e1b073132617e6" title="Click to view the MathML source">B_nclass="mathContainer hidden">class="mathCode">${\mathcal{B}}_n$ which has the strong N-property, improving mentioned Valdivia's theorem. Some applications to localization of bounded additive vector measures are provided.