V2EX measure-zero

Measure-zero

释义 Definition

(数学测度论)测度为零的;零测集的。 指在某种“测度”(最常见是勒贝格测度)下大小为 0 的集合。它们可能仍然包含无限多个点(例如一条线段上的单个点集、平面中的一条曲线),但在“长度/面积/体积”的意义上可忽略不计。(在概率语境中常对应“概率为 0”,但两者不总是完全等同。)

发音 Pronunciation (IPA)

/mr zro/

例句 Examples

The set of rational numbers in \[0,1\] has measure zero.
区间 \[0,1\] 内的有理数集合是测度为零的。

Although the curve contains infinitely many points, it is measure-zero in the plane, so it contributes no area.
虽然这条曲线包含无限多个点,但在平面中它是测度为零的,因此不会贡献任何面积。

词源 Etymology

measure(测度)源自拉丁语 mensura(“测量”),zero(零)来自意大利语 zero(与“空/无”相关)。短语 measure zero 出自 1920 世纪发展起来的测度论(与勒贝格积分密切相关),用于精确表达“在测度意义下可忽略”的集合;因此也常见于现代概率论中,用来讨论“几乎处处”(almost everywhere)成立的结论。

相关词 Related Words

文学与名著用例 Literary Works

  • Real and Complex Analysis Walter Rudin
  • Real Analysis: Modern Techniques and Their Applications Gerald B. Folland
  • Probability and Measure Patrick Billingsley
  • Real Analysis H. L. Royden & P. M. Fitzpatrick
  • An Introduction to Measure Theory Terence Tao
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