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1 vote

### If the set $f^{-1}\{a\}$ is measurable for all $a$, then is $f$ a measurable function?

As pointed out by Martin R your argument is not correct. I will assume that all uncountable subsets of $\mathbb R$ have the same cardinality as $\mathbb R$. If $E$ is a non-measurable set then $E$ ...
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1 vote

### Exemple where tower property of conditional expectation is NOT verify

Take this with a grain of salt, as I am also a learner of Probability. Your example is correct in the design (take $F_{1,2}$ such that $F_1 \not\subseteq F_2, F_2 \not\subseteq F_1$), but is wrong in ...

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