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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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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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