I cannot detect the fallacy in the set of the following statements in my inconsistent notes:
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A sigma algebra is a set of the sets in the generating set closed under the set operations countable union, countable intersection, set difference, relative complement.
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A Borel set is a subset of $\mathbb{R}$ constructed from open and closed intervals in $\mathbb{R}$ by taking the operations countable union and intersection.
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The set of Lebesgue measurable sets on $\mathbb{R}$ is a sigma algebra generated from the open and closed intervals in $\mathbb{R}$.
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By the statements 2 and 3 every Lebesgue measurable set on $\mathbb{R}$ is a Borel set.
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There is a Lebesgue measurable set that is not Borel.
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By 4 and 5, falsity.
Which statements in my notes are not true and why?
Best Answer
The problem is statement (3).
The Lebesgue measure algebra is indeed a $\sigma$-algebra, but it is generated by completing the Borel $\sigma$-algebra with respect to the null sets ideal.
One can prove that there are only $2^{\aleph_0}$ Borel sets, but since the Cantor set is Borel, and of measure zero, every subset of the Cantor set is measurable. But then again the Cantor set has cardinality $2^{\aleph_0}$, so it has $2^{2^{\aleph_0}}$ subsets, all of which are Lebesgue measurable; and so most of them are not even Borel sets.