Typical behaviour of integrable functions at infinity

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• 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 L₁:=L₁([0,∞))$L_1:=L_1\left([0,\infty )\right)$ and let C(1)$C\left(1\right)$ be the (Polish) space of nonnegative continuous functions 34b" title="Click to view the MathML source">f$f$ on 4b15f" title="Click to view the MathML source">[0,∞)$[0,\infty )$ such that ∫_[0,∞)f≤1${\int }_{[0,\infty )}f\le 1$, with the metric of uniform convergence on every compact subset of 4b15f" title="Click to view the MathML source">[0,∞)$[0,\infty )$. Denote $c_0^{+}:=\{\left(b_n\right)\in c_0:b_n>0\text{for all}n\in \mathbb{N}\}$. Then, for Y:=L₁$Y:=L_1$, the sets are comeagre of type b180553ef3a07de415120f0b069a1a" title="Click to view the MathML source">G_δ$G_{\delta }$. If Y:=C(1)$Y:=C\left(1\right)$, the analogous sets, with the phrase “for almost all” replaced by “for all”, are also comeagre of type b180553ef3a07de415120f0b069a1a" title="Click to view the MathML source">G_δ$G_{\delta }$.