Let ( Ω , A , μ ) be a measure space and f : Ω → R a measure nonnegative function that is summable against μ.It is known that ν ( A ) = ∫ A f μis also a measure on A . I have to prove that ∫ Ω g ν = ∫ Ω g f μfor all functions g : Ω → R that are summable against ν.I don't quite understand the notation, though. Specifically, on the proof statement, what does g f μ mean? Am I taking the composition of g and f? How does one even evaluate ∫ Ω g ν?Thanks in advance for any help.

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Let   (  Ω  ,      A    ,  μ  ) be a measure space and   f  :  Ω  →      R   a measure nonnegative function that is summable against   μ.It is known that  ν  (  A  )  =      ∫          A        f  μis also a measure on       A  . I have to prove that      ∫          Ω        g  ν  =      ∫          Ω        g  f  μfor all functions   g  :  Ω  →      R   that are summable against   ν.I don&#039;t quite understand the notation, though. Specifically, on the proof statement, what does   g  f  μ mean? Am I taking the composition of   g and   f? How does one even evaluate       ∫          Ω        g  ν?Thanks in advance for any help.

kovilovop2 · Accepted Answer

Regarding your questions on the notation   ν  (  A  )  =  ∫  f  d  μ and   ∫  g  d  ν  =  ∫  f  g  d  μ, and in particular   f  g means multiplication of   f and   g.The question you are asking can be answered using Radon Nikodym theorem. However, you can also directly approach this question by assuming first that   g is a simple function. Then using the fact that you can approximate any measurable function with simple functions and dominated convergence theorem you can generalize it to any function.

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