一个幽灵般的问题

[Physics-Lee]

[WENYUEHu03]

数学家首先考虑如何写出这个问题的样本空间？

[Physics-Lee]

可以啊，写呗。

[Yue9Shang]

我感觉这个对于这个数学问题，结果应该是1的。但考虑到神经元是有一定大小的，用这个数学模型并不合适，可能需要加上限制，红色点的区域应至少有一个长度。如果再假设该系统空间平移对称，温老师的模型是很有道理的。

[Physics-Lee]

@Yue9Shang

物理上，温老师的做法似乎是唯一一种可行的做法。

数学上，这个问题的答案很反直觉——把一条有限长的线段上任意有限长度染色，它上面任何其它有限长线段含有红色的点的概率竟然都是1。

[forrestjhliu]

According to measure theory, in Lebesgue measure, $\mu (\mathbb Q)= 0$, but every interval on [0,1] 's intersection with $\mathbb Q$ is not empty. So this is related to what is “线段” in the question. In math, you can easily fill up [0,1] by $\mathbb Q$ with zero measure, but in the real-world everything must have a "measure" >0.

[Physics-Lee]

Yeah, that is right, if you make $\mathbb Q$ to be red, then every interval with finite measure on [0,1] contains red points.

I am asking another math question: if you turn a finite interval (set it as $a$) of [0,1] to be red, the interval can be splitted infinitely, then what is the probability of another finite interval containing red points?

It seems that, no matter how small $a$ is, as long as $a > 0$, then $P = 1$