论文标题
在$ l^1 $上，连续功能的发散
On the $L^1$ and pointwise divergence of continuous functions
论文作者
论文摘要
对于一个连续功能的家族$ f_1，f_2，\ dots \ colon i \ to \ mathbb {r} $（$ i $是一个固定的间隔），带有$ f_1 \ le f_2 \ le f_2 \ le \ dots $定义了一个集合 $ i_f：= \ big \ {x \ in I \ colon \ lim_ {n \ to \ infty} f_n（x）=+\ infty \ big \ \}。 \ infty} \ int_x^y f_n（t）\：dt =+\ infty \ quad \ quad \ text {for All} x，y \ in I \ text {with} x <y。$$ 这个问题的起源是quasiarithmetic手段的极限行为。
For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ We study the properties of the family of all admissible $I_f$-s and the family of all admissible $I_f$-s under the additional assumption $$ \lim_{n \to \infty} \int_x^y f_n(t)\:dt=+\infty \quad \text{ for all }x,y \in I\text{ with }x<y.$$ The origin of this problem is the limit behaviour of quasiarithmetic means.
