It is my understanding that σ-algebras are necessary to satisfy certain desired properties of a measure, and that these conditions are mutually inconsistent if we consider arbitrary open sets of R . However, I have also read that power set of a set is necessarily a σ-algebra, so how come we cannot use the power set of R to define a measure space? I feel I am missing something fundamental here.

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It is my understanding that   σ-algebras are necessary to satisfy certain desired properties of a measure, and that these conditions are mutually inconsistent if we consider arbitrary open sets of       R  . However, I have also read that power set of a set is necessarily a   σ-algebra, so how come we cannot use the power set of       R   to define a measure space? I feel I am missing something fundamental here.

odmeravan5c · Accepted Answer

There are disjoint subsets   A and   B of       R   such that  μ  (  A  ∪  B  )  &amp;lt;  μ  (  A  )  +  μ  (  B  )violating the additivity condition of a measure μ induced from the &quot;length&quot; function on intervals. The collection of (Lebesgue) measurable sets is strictly smaller than the powerset.EDIT: It is pointed out in the comments that the powerset is an adequate sigma-algebra for other measures like &quot;atomic&quot; measures on       R  . Thus your intuition is not far off.

It is my understanding that σ -algebras are necessary to satisfy certain desir

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