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The question is to give and example of Lebesgue measurable set but not Borel measurable.
I know there exists subset of Cantor set that is not Borel measurable, since the cardinality of all Borel sets in $[0,1]$ is $\aleph_1$ while the cardinality of the subsets of Cantor set (defined on $[0,1]$) is $\aleph_2$, so it must contains subsets that are non-Borel measurable.
The problem is, the "cardinality of all Borel sets in $[0,1]$ is $\aleph_1$" requires something that is beyond the scope of first-year graduate level real analysis course. I cannot prove it using what I learned.
Can anyone give another example? Thank you!