陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Non-measurable sets via non-standard analysis」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Last year on this blog , I sketched out a non-rigorous probabilistic argument justifying the following well-known theorem:

Theorem 1. (Non-measurable sets exist) There exists a subset of the unit interval which is not Lebesgue-measurable .

已知结果和反例

The idea was to let E be a “random” subset of . If one (non-rigorously) applies the law of large numbers , one expects E to have “density” 1/2 with respect to every subinterval of , which would contradict the Lebesgue differentiation theorem .

I was recently asked whether I could in fact make the above argument rigorous. This turned out to be more difficult than I had anticipated, due to some technicalities in trying to make the concept of a random subset of (which requires an uncountable number of “coin flips” to generate) both rigorous and useful. However, there is a simpler variant of the above argument which can be made rigorous. Instead of letting E be a “random” subset of , one takes E to be an “alternating”

证明或构造的主线

Of course, in the standard model of the real numbers, it makes no sense to talk about “every other” or “every second” real number, as the real numbers are not discrete. If however one employs the language of non-standard analysis , then it is possible to make the above argument rigorous, and this is the purpose of my post today. I will assume some basic familiarity with non-standard analysis, for instance as discussed in this earlier post of mine .

We begin by selecting a non-principal ultrafilter and use it to construct non-standard models of the natural numbers and the unit interval by the usual ultrapower construction. We then let be an unlimited non-standard number, i.e. a non-standard natural number larger than any standard natural number. (For instance, one could take N to be the equivalence class in of the sequence .)

阅读时建议盯住的点

We can partition the non-standard unit interval into (non-standard) intervals for (the overlap of these intervals will have a negligible impact in our analysis). We then define the non-standard set to be the union of those with j odd; this is the formalisation of the idea of “every other real number” in the introduction. The key property about *E that we need here is the following symmetry property: if is any standard dyadic interval , then the (non-standard counterpart to th

We now return to the standard world, and introduce the standard set , defined as the collection of all standard whose non-standard representative lies in . This is certainly a standard subset of . We claim that it is not Lebesgue measurable, thus establishing Theorem 1. To see this, recall for any standard dyadic interval , that every non-standard irrational element of I lies in exactly one of *E or . Applying the transfer principle, we conclude that every standard irrational

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源