In the context of a measure space ( X , M , μ ), suppose f is a bounded measurable function with a ≤ f ( x ) ≤ b for μ-a.e. x ∈ X. Prove that for each integrable function g, there exists a number c ∈ [ a , b ] such that ∫ X f | g | d μ = c ∫ X | g | d μI tried to use the Intermediate value theorem for integral of Riemann but i had no idea. Somebody have any tip?

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In the context of a measure space   (  X  ,  M  ,      μ    ), suppose   f is a bounded measurable function with   a  ≤  f  (  x  )  ≤  b for   μ-a.e.   x  ∈  X. Prove that for each integrable function   g, there exists a number   c  ∈  [  a  ,  b  ] such that       ∫    X    f      |    g      |    d  μ  =  c      ∫    X        |    g      |    d  μI tried to use the Intermediate value theorem for integral of Riemann but i had no idea. Somebody have any tip?

Carleigh Shaffer · Accepted Answer

You don&#039;t need measure theory is an easy use of intermediate theorem for real valued functions…Consider  h  (  c  )  =  c      ∫    X        |    g      |    d  μ  −      ∫    X    f      |    g      |    d  μThen   h  (  a  )  ≤  0  ,  h  (  b  )  ≥  0, hence there is a   c  ∈  [  a  ,  b  ] s.t.   h  (  c  )  =  0 what gives you the claim.

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