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Let $f$ and $g$ be Lebesgue measurable nonnegative functions on $\mathbb{R}$. Let $A_y=\{x:f(x) \geq y\}$ Let $F(y)=\int_{A_y} g(x)dx$. Prove $\int_{-\infty}^\infty f(x)g(x)dx=\int_0^\infty F(y)dy$. I know this has to do with Fubini's theorem but I cannot prove it.

john
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    Hint: if $H(x,y) = 1$ when $f(x) \ge y$ and $0$ otherwise, $$\int_0^\infty F(y)\ dy = \int_0^\infty \int_0^\infty H(x,y) g(x) \ dx\ dy$$ What is $\int_0^\infty H(x,y)\ dy$? – Robert Israel May 25 '12 at 20:22
  • You might try to read again Robert's hint, s-l-o-w-l-y. – Did May 25 '12 at 20:31
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    Say $f(x)=7$. What are $H(x,0)$, $H(x,2)$, $H(x,6)$ and $H(x,9)$? What is the function $y\mapsto H(x,y)$? Now what is $\int_0^\infty H(x,y)dy$? – Did May 25 '12 at 20:40
  • Is it f(x)? I think I see it – john May 25 '12 at 20:43
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    If you see it (which I hope), you might want to write yourself a solution and to post it here. After a while, you may even accept it... :-) – Did May 25 '12 at 20:45
  • thank you. I got a solution – john May 25 '12 at 20:47

1 Answers1

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recall Fubini’s theorem and apply to couple of function $(f(x|t) ,p(t))$

robjohn
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user76143
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    Welcome to MSE. It is not clear how your answer will help solve the problem. Adding in more detail would be helpful. – Daryl May 05 '13 at 11:24