Let w : <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N </mrow>

Armeninilu 2022-07-01 Answered
Let w : N [ 0 , ) continuous.
For each f : N C such that n = 1 w ( n ) f ( n ) is absolutely convergent we define Λ f = n = 1 w ( n ) f ( n )
It is easy to prove that Λ is linear and satisfies that f : f ( N ) [ 0 , ) Λ f [ 0 , )
(I think this last property has a name but I don't know what it is)
By the Riesz representation theorem there is only one positive measure ν such that Λ f = N f d ν . How can I find the measure ν that fulfills this property?
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Answers (2)

Stevinivm
Answered 2022-07-02 Author has 18 answers
If μ is the counting measure, consider ν defined by d ν = w d μ. This is a positive measure since w 0. Thus f d ν = f w d μ = n f ( n ) w ( n )

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George Bray
Answered 2022-07-03 Author has 12 answers
Compute for E N
ν ( E ) = 1 E d ν = Λ ( 1 E ) = n w ( n ) 1 E ( n ) = n E w ( n ) .

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