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Not sure if this is the right place to post such a rookie question, but I'd appreciate some quick clarification. Is there such a term as a "Borel measurable set"? I've seen it used all over the place but I'm pretty sure it simply means a "Borel set" most of the time, which does not necessarily invoke the notion of a measure. Is this a well-defined term and if so, how is it different from a "Borel set"?

J. W. Tanner
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Bunbury
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    As far as I can tell these would refer to the same thing. – Math1000 May 01 '19 at 17:08
  • If you take the family of open intervals and make countable union countable intersection and consider the complement for each such subsets that a member from this family is called a Borel set it contains open closed intervals G$-\delta$ , F$-\sigma$ sets. It is what we call is the smallest sigma algebra containing the open interval – IrbidMath May 01 '19 at 17:21
  • Those sets are called Borel measurable sets. If you define the outer measure then consider sets $E$ which statisfies $m(A)=m(A\cap E)+m(A\cap E^c)) $ for any $A\in X$ then we call $E$ a measurable set. An example of measurable set which is not Borel is the cantor set – IrbidMath May 01 '19 at 17:25
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    @Ameryr Cantor set is closed, thus Borel. – mihaild May 01 '19 at 18:02
  • @mihaild my bad I thought it is not it uncountable with measure zero . Is there any examples of measurable but not Boreal? https://math.stackexchange.com/questions/141017/lebesgue-measurable-set-that-is-not-a-borel-measurable-set – IrbidMath May 01 '19 at 19:11
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    There are - there are just continuum many Borel sets, but all subsets of Cantor set is measurable, so there are more then continuum measurable sets. – mihaild May 01 '19 at 19:16

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Yes, "Borel measurable" normally means the same thing as "Borel".

This is related to terminology from abstract measure theory, where you have a set $X$ equipped with a $\sigma$-algebra $\mathcal{F} \subset \mathcal{P}(X)$. If $A \in \mathcal{F}$, we commonly say that $A$ is "measurable" (if it's understood from context what $\sigma$-algebra we are using) or "$\mathcal{F}$-measurable" (if it's not). So "Borel measurable" is like saying "$\mathcal{F}$-measurable, where $\mathcal{F}$ is the Borel $\sigma$-field".

Nate Eldredge
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