I am reading the example of "Boreal measure is not complete" from Wiki:
https://en.wikipedia.org/wiki/Complete_measure
In the first example, it says
The power set of the Cantor set has cardinality strictly greater than that of the reals. Thus there is a subset of the Cantor set that is not contained in the Borel sets.
My questions are the following:
- Why "Cantor set has cardinality strictly greater than that of the reals"?
- Why "there is a subset of the Cantor set that is not contained in the Borel sets"? Can anyone give me an example of this?
Thanks!
As mentioned in the comments, the Cantor set has cardinality $2^{\aleph_0}$ (i.e. the same cardinality as the reals). Thus its power set has cardinality $2^{2^{\aleph_0}},$ which by Cantor's theorem is greater than the cardinality of the reals. On the other hand, it can be shown that there are only $2^{\aleph_0}$-many Borel sets (this isn't trivial). Thus, there are more subsets of the Cantor set than there are Borel sets, so there is a subset of the Cantor set that is not a Borel set.