Here is an excercise of measure theory and real analysis. can you help me with it?Let (X,M, μ) be a measure space and f : X → [ 0 , ∞ ] a measurable function. Prove that ∫ X f d μ &gt; 0 ⇔ μ ( { x ∈ X ∣ f ( x ) &gt; 0 } ) &gt; 0.I tried to use the definition of the integral by the approximation theorem of measurable functions by simple functions, but I haven't been able to come to the conclusion.

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Here is an excercise of measure theory and real analysis. can you help me with it?Let (X,M,  μ) be a measure space and   f  :  X  →  [  0  ,  ∞  ] a measurable function. Prove that      ∫          X        f  d  μ  &amp;gt;  0    ⇔    μ            (            {    x    ∈    X    ∣    f    (    x    )    &amp;gt;    0    }            )        &amp;gt;  0.I tried to use the definition of the integral by the approximation theorem of measurable functions by simple functions, but I haven&#039;t been able to come to the conclusion.

Jake Mcpherson · Accepted Answer

Let   A  =  {  x  ∣  f  (  x  )  &amp;gt;  0  } and       A          n        =  {  x  ∣  f  (  x  )  &amp;gt;      1    n    }. Note that   A  =      ∪          n      =      1              ∞            A          n      . If   μ  (  A  )  &amp;gt;  0, then   μ  (      A          n        )  &amp;gt;  0 for some   n. Therefore   ∫  f  ≥      ∫                  A                  n                      f  ≥      1    n    μ  (      A          n        )  &amp;gt;  0.Conversely, suppose that   ∫  f  &amp;gt;  0. Denote   B  =  {  x  ∣  f  (  x  )  =  0  }. We have that   ∫  f  =      ∫          A        f  +      ∫          B        f  =      ∫          A        f. If   μ  (  A  )  =  0, then       ∫          A        f  =  0 and we arrive a contradiction. Therefore   μ  (  A  )  &amp;gt;  0 (By definition,      ∫          A        f  =  ∫      1          A        f and the integrand       1          A        f  =  0 a.e. if   μ  (  A  )  =  0. I hope that you would not ask me to prove that   ∫  g  =  0 whenever   g  =  0 a.e.)

Here is an excercise of measure theory and real analysis. can you help me with it? Let (X,M, &#

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