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?

Question

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&#039;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?

Stevinivm · Accepted Answer

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  )

George Bray · Accepted Answer

Compute for   E  ⊆      N    ν  (  E  )  =  ∫            1        E    d  ν  =  Λ  (            1        E    )  =      ∑    n    w  (  n  )            1        E    (  n  )  =      ∑          n      ∈      E        w  (  n  )  .

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

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