Suppose that ν &lt;&lt; μ. Then we can find a non-negative f s.t. ν ( E ) = ∫ E d ν = ∫ E f d μSo far, things seem clear to me. My question is the following: Though it makes intuitive sense, how can we be sure that ∫ E d ν = ∫ E f d μ ⇒ ∫ E g d ν = ∫ E g f d μfor all integrable functions g? In the top answer to the question I linked to above, the following claim is made:For every integrable g, the following formula holds: ∫ E g d ( ∫ E f d μ ) = ∫ E g f d μIt therefore seems that a justification/proof of this claim would answer my question.I have been exposed to some measure theory and integration theory a few years back, and as I was revising some material recently, this claim was not clear to me. Perhaps this claim is obvious, and my confusion simply arises from a poor understanding of important definitions. Either way, any help in understanding this claim is much appreciated.

Question

Suppose that   ν  &amp;lt;&amp;lt;  μ. Then we can find a non-negative f s.t.  ν  (  E  )  =      ∫          E        d  ν  =      ∫          E        f  d  μSo far, things seem clear to me. My question is the following: Though it makes intuitive sense, how can we be sure that      ∫          E        d  ν  =      ∫          E        f  d  μ  ⇒      ∫          E        g  d  ν  =      ∫          E        g  f  d  μfor all integrable functions g? In the top answer to the question I linked to above, the following claim is made:For every integrable g, the following formula holds:      ∫          E        g  d  (      ∫          E        f  d  μ  )  =      ∫          E        g  f  d  μIt therefore seems that a justification/proof of this claim would answer my question.I have been exposed to some measure theory and integration theory a few years back, and as I was revising some material recently, this claim was not clear to me. Perhaps this claim is obvious, and my confusion simply arises from a poor understanding of important definitions. Either way, any help in understanding this claim is much appreciated.

Jordan Mcpherson · Accepted Answer

The result is given as a theorem in the book Real Analysis: Measures, Integrals and Applications, by Makarov and Podkorytov (p. 146). The proof in that book is essentially the same as the argument given by Prahlad Vaidyanathan.

Suppose that ν << μ . Then we can find a non-negative

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