Examples of non-measurable sets in $\mathbb {R}$ As a $ \sigma $-algebra is by definition closed under a countable union, and as singletons in $ \mathbb {R} $ are Borel-measurable, it follows that a countable subset of $ \mathbb {R} $ is Borel-measurable and that $ S $, being a countable union of countable (hence Borel-measurable) subsets of $ \mathbb {R} $, is Borel-measurable
Lebesgue measurable set that is not a Borel measurable set In short: Is there a Lebesgue measurable set that is not Borel measurable? They are an order of magnitude apart so there should be plenty examples, but all I can find is "add a Lebesgue-zero measure set to a Borel measurable set such that it becomes non-Borel-measurable"
Intuition behind the Caratheodory’s Criterion of a measurable set The only explanation I've ever seen is that a set is measurable if it 'breaks up' other sets in the way you'd want I don't really see why this is the motivation though One reason I am not comfortable with it is that you require a measurable set to break up sets which, according to this definition, are non-measurable; why would you require that?
general topology - What makes the elements of sigma algebra measurable . . . Is it an implication of the definition? If yes, how is it avoiding admitting non-measurable sets into sigma algebra? When they say measurable non-measurable, what is the measure they are talking about? Lebesgue, counting, probability? It seems there is an implicit measure every time someone says a set is measurable non-measurable
Relationship Between Borel and Lebesgue Measurable Sets I'm currently going through Real Analysis by Stein and Shakarchi On page 23 of the book, Stein claims that the set of all Lebesgue measurable sets can be given by adjoining all subsets of Borel se