In 1905 Lebesgue showed that there is a sequence of continuous functions, put $f_{n}:[0,1]longrightarrow [0,1]$, which interpolates any sequence in $[0,1]$, that is, given $(a_{n})_{ngeq 1}subset [0,1]$ there is $tin [0,1]$ such that $f_{n}(t)=a_{n}$ for each positive integer $n$. This result was improved (in the sense of Theorem ) in 1998 by Y. Benyamini. In this paper, we generalize the Benyamini's result in Theorem . The key tool for this goal are the so called $lpha $-dense curves. We apply our results to approach the solution of a certain infinite-dimensional linear program with a countable number of constraints.
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机译：1905年，勒贝格（Lebesgue）证明了一个连续函数的序列，将$ f_ {n}：[0,1] longrightarrow [0,1] $插入其中，可以对$ [0,1] $中的任何序列进行插值，即，给定$（a_ {n}）_ {n geq 1} subset [0,1] $，在[0,1] $中有$ t 使得$ f_ {n}（t）= a_ {n}每个正整数$ n $。 Y. Benyamini在1998年（从定理的意义上）改善了这一结果。在本文中，我们在定理中推广了Benyamini的结果。实现此目标的关键工具是所谓的$ alpha $-密集曲线。我们将我们的结果应用于具有可数约束的某个无限维线性程序的求解。
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