Timeline for A possible measure-theoretic pathology
Current License: CC BY-SA 4.0
6 events
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| Sep 22, 2023 at 20:40 | comment | added | Mateusz Kwaśnicki | Indeed, the definition of $\psi$ was wrong. Your construction of $\mu$ and $V$ is very elegant and explicit! There are of course many other ways to prove existence of $\mu$ and $V$. | |
| Sep 22, 2023 at 20:37 | history | edited | Mateusz Kwaśnicki | CC BY-SA 4.0 |
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| Sep 22, 2023 at 17:16 | comment | added | pmw | I'm a little confused about the statement "μ(A)=|ϕ−1(A)|=|ψ(A)| for every A=(0,x)"; don't we have μ(A)=ϕ(x) =|ϕ(A)| by definition of ϕ? Nonetheless, I think this construction of W is correct, so thank you, Mateusz! A simple example of a singular, atomless, full-support probability measure on [0,1] is obtained by i.i.d. drawings of bits from {0,1} with probabilities 2/3 and 1/3, respectively, to obtain the binary expansion of a real number. Then V could be the set of reals whose binary expansions contain, in the limit, twice as many zeroes as ones. | |
| Sep 16, 2023 at 0:13 | history | edited | Mateusz Kwaśnicki | CC BY-SA 4.0 |
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| Sep 16, 2023 at 0:08 | history | edited | Mateusz Kwaśnicki | CC BY-SA 4.0 |
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| Sep 15, 2023 at 23:58 | history | answered | Mateusz Kwaśnicki | CC BY-SA 4.0 |