# Why bring in Borel sets and $\sigma$-algebra in probability theory?

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?

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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 $[a+r,b+r]$).
This is why we work with $\sigma$-algebras, and the Borel one is very suitable because it is generated by the subintervals.