There is a conceptual puzzle that I don't understand about nonmeasurable sets. Take the unit interval [itex][0, 1][/itex] and let [itex]S[/itex] be some subset. Now, generate (using a flat distribution) a sequence of reals in the interval:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]r_1, r_2, r_3, ...[/itex]

Then we can define the relative frequency up to [itex]n[/itex] as follows:

[itex]f_n = 1/n \sum_j \Pi_S(r_j)[/itex]

where [itex]\Pi_S(x)[/itex] is the characteristic function for [itex]S[/itex]: it returns 1 if [itex]x \in S[/itex] and 0 otherwise.

If [itex]S[/itex] is measurable, then with probability 1, [itex]lim_{n \rightarrow \infty} f_n = \mu(S)[/itex], where [itex]\mu(S)[/itex] is the measure of [itex]S[/itex]. But if [itex]S[/itex] is not measurable, then what will be true about [itex]f_n[/itex]? Will it simply not have a limit? Or since this is a random sequence, we can ask what is the probability that it will have a limit. Does that probability exist?

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# A Relative frequency and nonmeasurable sets

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