Abstract

S. Marcus used a fat Cantor set to show that there is a quasicontinuous function from the interval $[0,1]$ to $\mathbb{R}$ that is not Lebesgue measurable. However, his article is not very well known. We give a very simple example of a quasicontinuous non-Lebesgue measurable function from $[0,1]$ to $\mathbb{R}$ and we show that there are $2^{\mathfrak{c}}$ quasicontinuous non-Lebesgue measurable functions and $2^{\mathfrak{c}}$ quasicontinuous Lebesgue measurable functions.

中文翻译：

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有2c个拟连续的非Lebesgue可测量函数

摘要

S. Marcus使用一个胖Cantor集证明该区间存在拟连续函数 $[0，\mathrm{1个}]$ 至 $\mathbb{[R}$Lebesgue无法测量。但是，他的文章不是很出名。我们给出一个非常连续的非莱贝格可测函数的简单例子$[0，\mathrm{1个}]$ 至 $\mathbb{[R}$ 我们证明有 ${\mathrm{2个}}^{\mathfrak{C}}$ 拟连续非莱贝格可测函数和 ${\mathrm{2个}}^{\mathfrak{C}}$ 准连续Lebesgue可测量的功能。