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I am a student of Mathematics.I am self-studying measure theory.I have defined measurable set with respect to an outer measure $m^*$ as a set $A$ such that $m^*(E)=m^*(E\cap A)+m^*(E\cap A^c)$ for each set $E$.Measurable set with respect to the Lebesgue outer measure are called Lebesgue measurable sets.We know Borel sets are Lebesgue measurable and also know that sets with outer measure zero are Lebesgue measurable.Now I want to know if Lebesgue measurable sets is the smallest $\sigma$-algebra containing Borel sets and sets with outer measure $0$ and if so,how to prove it?Can someone help me with this? And if this is so,then is it true that Lebesgue measurable sets are generated by taking set difference between Borel sets and outer measure zero sets?

Kishalay Sarkar
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  • Yes, in fact, the $\sigma -$algebra of Leabesgue measurable set is the completion of the Borel $\sigma -$algebra. To prove it, one can prove that eavry measurable set (in lebesgue sense) car be written as $B\cup N$ where $B$ is a Borel set and $N$ a null set. – Surb Mar 29 '22 at 12:54
  • @Surb Just write explicitly how to show that Leesgue measurable sets form the smallest $\sigma$-algebra containing Borel sets and null sets. – Kishalay Sarkar Mar 29 '22 at 12:59
  • @KishalaySarkar https://math.stackexchange.com/questions/3420145/lebesgue-measurable-set-union-of-borel-set-and-null-set – Gono Mar 29 '22 at 13:40
  • In fact, you only need to use the "tiniest portion" of the Borel sets. The Lebesgue measurable sets are the smallest $\sigma$-algebra containing the open sets and all sets of outer Lebesgue measure zero. Clearly, "open" can also be replaced by "closed". In fact, if we're in the real line, then "open" can be replaced by the much smaller and more concrete collection consisting of all open intervals with rational endpoints. – Dave L. Renfro Mar 29 '22 at 13:52
  • @DaveL.Renfro Please write a proof for me.You have to prove the fact that a Lebesgue measurable set is the union of a Borel set and a null set using the definition of Lebesgue measurable set $A$ which is $m^(E)=m^(E\cap A)+m^*(E\cap A^c)$ for any set $E\subset \mathbb R$ – Kishalay Sarkar Mar 29 '22 at 13:55
  • I don't have time now (nor, frankly, the desire) to develop the theory from this beginning until the result you're interested in shows up, especially since it can be found in many (hundreds of, in fact) standard texts. A useful word for looking in the indexes of books, or for googling, is regularity (or regular). In MSE, see Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero AND Equivalent definitions of Lebesgue measurability. – Dave L. Renfro Mar 29 '22 at 14:12

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