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I’m learning measure theory and a question came up and struck me. I know that Borel sigma algebra on the real number set is generated by a topology set of open intervals. My question is that is it required that topology must contains all of the open intervals in the real number set? Or any set of some open intervals can be used to generate the Borel sigma algebra on real number set? In other words, is THE Borel sigma algebra on real set unique?

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  • $\begingroup$ Certainly for any subset the set of open intervals in $\mathbb R$ (or more generally, any subset of the power set of $\mathbb R$), there is a unique generated $\sigma$-algebra. But as far as I know, the term "Borel" $\sigma$-algebra is reserved to mean the $\sigma$-algebra generated by the set of all open intervals in $\mathbb R$, and yes, it is unique. $\endgroup$ – Bungo Oct 30 '17 at 3:45
  • $\begingroup$ Yes, it is unique in the sense that it is defined as generated by all open sets. $\endgroup$ – amsmath Oct 30 '17 at 3:46
  • $\begingroup$ If, for example, you would generate your sigma-algebra by all open sets in $(0,\infty)$, you would get all Borel sets in $(0,\infty)$ together with their unions with $(-\infty,0]$. $\endgroup$ – amsmath Oct 30 '17 at 3:50
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First you learn what an open interval is, in say high school or possibly sooner. You define these as sets of the form $(a,b)$ where $a<b$. Then in a topology course you define what a $topology$ is.... namely a collection of objects with certain properties and we call these objects "open sets". It just happens that the way we define the Borel Sigma Algebra is that it is the topology that is generated by the open intervals (where "open" here refers, first and foremost, to your naive high school definition of open). Obviously, though, because you have defined your basis as these open intervals, it necessarily follows that they are open sets.... in THIS topology. You could have just as easily defined closed intervals as open sets, what sort of topology would this generate? Check for yourself. There is more than one way to generate the Borel sigma algebra but yes there really is only one.

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