1. ## Measure Theory problem

I know that the cardinality of the power set of R is strictly greater than c, and that the cardinality of the Borel sigma algebra on R is equal to c.

I'm supposed to use this to show that there is a Lebesgue measurable subset of R that is not a Borel set.

I'm a bit confused, because it seems like to do this I would have to show that there are Lebesgue measurable sets that are not Borel sets anyway apart from the cardinality argument, since not every subset of R is Lebesgue measurable.

2. Originally Posted by Diamondlance
I know that the cardinality of the power set of R is strictly greater than c, and that the cardinality of the Borel sigma algebra on R is equal to c.

I'm supposed to use this to show that there is a Lebesgue measurable subset of R that is not a Borel set.

I'm a bit confused, because it seems like to do this I would have to show that there are Lebesgue measurable sets that are not Borel sets anyway apart from the cardinality argument, since not every subset of R is Lebesgue measurable.

The idea is to show that the cardinality of the set of the Lebesgue measurable subsets of $\mathbb{R}$ equals the cardinality of the power set of $\mathbb{R}$. Because of what you wrote, this would be enough.
This results from the following fact: there exists a negligible uncountable Borel subset $C$ of $\mathbb{R}$. More precisely, its cardinality is that of $\mathbb{R}$.
Supposing this is true, then every subset of $C$ is a Lebesgue subset (because they're included in a negligible Borel subset). However, because $|C|=|\mathbb{R}|$, the set of the subsets (i.e. the power set) of $C$ is in bijection with the power set of $\mathbb{R}$. This proves what we want.
Coming back to the announced fact, the usual example is a Cantor subset, and I'd bet you've already seen this. For instance, it can be the set of the numbers in $[0,1]$ such that all of their decimals (in base 10) are even. Or (the usual one) the set of the numbers in $[0,1]$ such that the only numbers appearing in their representation in base 3 are 0 and 2.