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I am trying to find a construction of a measurable subset that is not Borel, and ask

for a ref. in this argument ( see the ***) used to show the existence of such sets:

i) Every set of outer measure 0 is measurable, since:

0=m^{*}(S)≥m_{*}(S) , forcing equality.

ii) Every subset of the Cantor set is measurable, by i), and there are 2^{c}=2^{2}^{Aleph_0}such subsets.

iii)*** The process of producing the Fσ , G_{δ}, F_{σδ},.....

produces only 2^{Aleph_0}sets. ***

iv) Since the 2 cardinalities are different, there must be a set as described in ii), i.e., a Lebesgue-measurable set that is not Borel.

So, questions:

1)How do we show the process in iii) produces only c sets.

2)Anyone know of an actual construction of this set?

Thanks.

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# Lebesgue Measurable but not Borel sets.

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