Consider the set of natural numbers N with the σ-algebra P ( N ) and the count measure μ. If f : N → [ 0 , ∞ ), then ∫ f d μ = ∑ n = 1 ∞ f ( n ) .I feel that I need to prove at first that f is a measurable function and then define a sequence f n ∈ L + s. t. f = ∑ n f n , but I have no idea if it's the right way nor how to do it. I need help.Note: I can use any theorem, no restrictions.

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Consider the set of natural numbers       N   with the   σ-algebra   P  (      N    ) and the count measure   μ. If   f  :      N    →  [  0  ,  ∞  ), then  ∫  f  d  μ  =      ∑          n      =      1              ∞        f  (  n  )  .I feel that I need to prove at first that f is a measurable function and then define a sequence             f      n        ∈      L    +   s. t.   f  =      ∑          n            f    n  , but I have no idea if it&#039;s the right way nor how to do it. I need help.Note: I can use any theorem, no restrictions.

delalbaef · Accepted Answer

Since the   σ-algebra in the domain is   P  (      N    ), all functions   f  :      N    →      R   are measurable.Given   f  :      N    →  [  0  ,  ∞  ). For each   k  ∈      N  , let       E    k    =      N    ∩  [  0  ,  k  ] and       f    k    =  f  ⋅      χ                  E        k            . Then,       f    k   are simple functions,       f    k    ↗  f and   ∫      f    k    d  μ  =      ∑          n      =      1              k        f  (  n  ). So, we have,  ∫  f  d  μ  =      lim          k      →      ∞        ∫      f    k    d  μ  =      lim          k      →      ∞            ∑          n      =      1              k        f  (  n  )  =      ∑          n      =      1              ∞        f  (  n  )  .

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