Typical behaviour of integrable functions at infinity

详细信息    查看全文

• 作者：Marek Balcerzak ^{marek.balcerzak@p.lodz.pl" class="auth_mail" title="E-mail the corresponding author} ; Adam Majchrzycki ^{majchrzu@wp.pl" class="auth_mail" title="E-mail the corresponding author} ; Filip Strobin ; ^{filip.strobin@p.lodz.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Lebesgue integrable functions ; Baire category ; Lesigne function
• 刊名：Indagationes Mathematicae
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：27
• 期：4
• 页码：893-901
• 全文大小：202 K

文摘

We obtain the following extension of a theorem due to Lesigne. Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si1.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=be80d2b730443942e293ff1870f3a8ac" title="Click to view the MathML source">L₁:=L₁([0,∞))class="mathContainer hidden">class="mathCode">$L_1:=L_1\left([0,\infty )\right)$ and let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si2.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=789498746437d07045267c2992a6d902" title="Click to view the MathML source">C(1)class="mathContainer hidden">class="mathCode">$C\left(1\right)$ be the (Polish) space of nonnegative continuous functions class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si3.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=6453705f4a36708fe2b235379e2e134b" title="Click to view the MathML source">fclass="mathContainer hidden">class="mathCode">$f$ on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si4.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=286bd533ce24df56fe155931c344b15f" title="Click to view the MathML source">[0,∞)class="mathContainer hidden">class="mathCode">$[0,\infty )$ such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si112.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=ca5dbd979a58265fe920e2965a535bf1" title="Click to view the MathML source">∫_[0,∞)f≤1class="mathContainer hidden">class="mathCode">${\int }_{[0,\infty )}f\le 1$, with the metric of uniform convergence on every compact subset of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si4.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=286bd533ce24df56fe155931c344b15f" title="Click to view the MathML source">[0,∞)class="mathContainer hidden">class="mathCode">$[0,\infty )$. Denote class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si7.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=06f7fa20637767d1f5b63d753338121c">class="imgLazyJSB inlineImage" height="17" width="223" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S001935771630009X-si7.gif">class="mathContainer hidden">class="mathCode">$c_0^{+}:=\{\left(b_n\right)\in c_0:b_n>0\text{for all}n\in \mathbb{N}\}$. Then, for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si8.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=58d4d0331517bfdad5fdd7b3c957e295" title="Click to view the MathML source">Y:=L₁class="mathContainer hidden">class="mathCode">$Y:=L_1$, the sets are comeagre of type class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si11.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=b3b180553ef3a07de415120f0b069a1a" title="Click to view the MathML source">G_δclass="mathContainer hidden">class="mathCode">$G_{\delta }$. If class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si12.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=b62888c164a4b355d5e8a4f5df2abf63" title="Click to view the MathML source">Y:=C(1)class="mathContainer hidden">class="mathCode">$Y:=C\left(1\right)$, the analogous sets, with the phrase “for almost all” replaced by “for all”, are also comeagre of type class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S001935771630009X&_mathId=si11.gif&_user=111111111&_pii=S001935771630009X&_rdoc=1&_issn=00193577&md5=b3b180553ef3a07de415120f0b069a1a" title="Click to view the MathML source">G_δclass="mathContainer hidden">class="mathCode">$G_{\delta }$.