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In Probability theory, we can simply use power set of the underlying sample space as the event space. Why go into higher concepts of Borel sets, $\sigma$-algebra and measure?

Is it just an instance of generalization or does it address some flaw in the use of power sets?

Thank you in advance :)

Asaf Karagila
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AdityaGhosh
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1 Answers1

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If you consider uniform probability on an uncountable space, e.g. the unit interval $[0,1]$ then obviously the probability should be invariant under translations (i.e. the probability of being in the subinterval $[a,b]$ is the same as being in the subinterval $[a+r,b+r]$).

It turns out that under the common axioms of mathematics (which include the axiom of choice) there are sets to which we cannot assign probabilities and require that the probability is countably additive.

This is why we work with $\sigma$-algebras, and the Borel one is very suitable because it is generated by the subintervals.

Asaf Karagila
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    To someone wanting to understand the bigger picture, this answer raises more questions than it anwsers. – Johannes Gerer Apr 04 '13 at 17:34
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    @user4514: That's a good thing. – Asaf Karagila Apr 04 '13 at 17:36
  • Sorry to bring this up after a decade, but are there really such sets that are of interest? Yes, Banach T paradox shows such sets exist, but every "useful" set is usually some union of finitely sized intervals and their products. It seems that if we constrained ourselves to intervals we wouldn't need that overhead. – Nathaniel Bubis Jan 17 '24 at 06:04