It is my understanding that &#x03C3;<!-- σ --> -algebras are necessary to satisfy certain desir

Semaj Christian 2022-06-25 Answered
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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Answers (1)

odmeravan5c
Answered 2022-06-26 Author has 20 answers
There are disjoint subsets A and B of R such that
μ ( A B ) < μ ( A ) + μ ( B )
violating the additivity condition of a measure μ induced from the "length" 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 "atomic" measures on R . Thus your intuition is not far off.

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My idea for a solution is to use Fatou's lemma:
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