- #1
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Hi, All:
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 2c=22Aleph_0 such subsets.
iii)*** The process of producing the Fσ , Gδ , Fσδ ,...
produces only 2Aleph_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.
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 2c=22Aleph_0 such subsets.
iii)*** The process of producing the Fσ , Gδ , Fσδ ,...
produces only 2Aleph_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.