Let X be a set, A a ring of subsets of X, μ ∶ A...

Shea Stuart

Shea Stuart

Answered

2022-07-09

Let X be a set, A a ring of subsets of X, μ A R ¯ 0 a premeasure and μ the outer measure generated by μ. (By Caratheodory)
If E M μ and satisfacies μ ( E ) < , then for each ε existx A A such that μ ( A E ) < ε
I try to test this, my first idea was to give a cover of E, I just don't know if I can find said cover in A , as the measure is not finite, so the extension is not unique, someone can give me a hint how to proceed?

Answer & Explanation

iskakanjulc

iskakanjulc

Expert

2022-07-10Added 18 answers

By definition of the outer measure, there is a countable collection { Q j } of elements in A such that
j = 1 μ ( Q j ) μ ( E ) + ε .
Since μ ( E ) < , the series above converges. Hence there is an N so that
j = N + 1 μ ( Q j ) < ε 2 .
Consider A = j = 1 N Q j .
Addison Trujillo

Addison Trujillo

Expert

2022-07-11Added 6 answers

Be careful, since μ ( E ) = inf { j = 1 μ ( Q j ) : { Q j } j = 1 A , E j = 1 Q j } is not , μ ( E ) + ε ( > μ ( E ) ) is not an lower bound for { j = 1 μ ( Q j ) : { Q j } j = 1 A , E j = 1 Q j }, and so there exists { Q j } j = 1 A with E j = 1 Q j such that j = 1 μ ( Q j ) < μ ( E ) + ε. In other words, the condition μ ( E ) < is necessary for this step.

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