Let $a_n$ be either 0 or 1, with 50% chance. Does $x_1=\sum_1^\infty a_n/2^n$ exists as a real number, actually as a random variable? If so, let $(x_n,y_n,z_n)$ be generated the same way in 3-space. Let K be non-measurable set in $[0,1]\times [0,1]\times [0,1]$, and $k_n$ be 1 iff $(x_n,y_n,z_n)$ is in K, otherwise $0$. So $k_n$ is a random variable, too. Set $P(K)=\lim (k_n/n)$, which is convergent with probability 1 to the same number. Is P(K) not a measure of K? It seems it is. What was wrong in my definition?
2026-02-22 21:03:04.1771794184
Measuring a non-measurable set with probability?
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