Suppose that $F,G in NBV$ and $-infty <a<b< infty$, how show that
Suppose that $F,G in NBV$ and $-infty <a<b< infty$, how show that
$displaystyleint_{[a,b]} dfrac{F(x)+F(x-)}{2}dG(x) + displaystyleint_{[a,b]} dfrac{G(x)+G(x-)}{2}dF(x) = F(b)G(b)-F(a-)G(a-)$
where $F(a-)=lim_{x to a^-} F(x) $.
This exercise is from Folland Real analysis. The author suggests that he imitate the proof of Theorem 3.36, but I am not able to use the same lines. Can someone give a tip?
measure-theory functions
asked Nov 21 '18 at 20:19
Ricardo Freire
392110
add a comment |
Suppose that $F,G in NBV$ and $-infty <a<b< infty$, how show that
$displaystyleint_{[a,b]} dfrac{F(x)+F(x-)}{2}dG(x) + displaystyleint_{[a,b]} dfrac{G(x)+G(x-)}{2}dF(x) = F(b)G(b)-F(a-)G(a-)$
where $F(a-)=lim_{x to a^-} F(x) $.
This exercise is from Folland Real analysis. The author suggests that he imitate the proof of Theorem 3.36, but I am not able to use the same lines. Can someone give a tip?
measure-theory functions
asked Nov 21 '18 at 20:19
Ricardo Freire
392110
Did you try approximating with functions in $C^1$ ?
– reuns
Nov 21 '18 at 21:48
I think this does not help. There probably must be a trick to use along with the fubini's theorem
– Ricardo Freire
Nov 21 '18 at 22:31
Fubini is obtained from approximating with functions in $C^0_c$. What fails when approximating with functions in $C^1$ ?
– reuns
Nov 22 '18 at 18:40
add a comment |
Suppose that $F,G in NBV$ and $-infty <a<b< infty$, how show that
$displaystyleint_{[a,b]} dfrac{F(x)+F(x-)}{2}dG(x) + displaystyleint_{[a,b]} dfrac{G(x)+G(x-)}{2}dF(x) = F(b)G(b)-F(a-)G(a-)$
where $F(a-)=lim_{x to a^-} F(x) $.
This exercise is from Folland Real analysis. The author suggests that he imitate the proof of Theorem 3.36, but I am not able to use the same lines. Can someone give a tip?
measure-theory functions
asked Nov 21 '18 at 20:19
Ricardo Freire
392110
Suppose that $F,G in NBV$ and $-infty <a<b< infty$, how show that
$displaystyleint_{[a,b]} dfrac{F(x)+F(x-)}{2}dG(x) + displaystyleint_{[a,b]} dfrac{G(x)+G(x-)}{2}dF(x) = F(b)G(b)-F(a-)G(a-)$
where $F(a-)=lim_{x to a^-} F(x) $.
This exercise is from Folland Real analysis. The author suggests that he imitate the proof of Theorem 3.36, but I am not able to use the same lines. Can someone give a tip?
measure-theory functions
measure-theory functions
asked Nov 21 '18 at 20:19
Ricardo Freire
392110
asked Nov 21 '18 at 20:19
Ricardo Freire
392110
asked Nov 21 '18 at 20:19
Ricardo Freire
392110
asked Nov 21 '18 at 20:19
Ricardo Freire
392110
asked Nov 21 '18 at 20:19
Ricardo Freire
392110
392110
Did you try approximating with functions in $C^1$ ?
– reuns
Nov 21 '18 at 21:48
I think this does not help. There probably must be a trick to use along with the fubini's theorem
– Ricardo Freire
Nov 21 '18 at 22:31
Fubini is obtained from approximating with functions in $C^0_c$. What fails when approximating with functions in $C^1$ ?
– reuns
Nov 22 '18 at 18:40
add a comment |
Did you try approximating with functions in $C^1$ ?
– reuns
Nov 21 '18 at 21:48
I think this does not help. There probably must be a trick to use along with the fubini's theorem
– Ricardo Freire
Nov 21 '18 at 22:31
Fubini is obtained from approximating with functions in $C^0_c$. What fails when approximating with functions in $C^1$ ?
– reuns
Nov 22 '18 at 18:40
Did you try approximating with functions in $C^1$ ?
– reuns
Nov 21 '18 at 21:48
Did you try approximating with functions in $C^1$ ?
– reuns
Nov 21 '18 at 21:48
I think this does not help. There probably must be a trick to use along with the fubini's theorem
– Ricardo Freire
Nov 21 '18 at 22:31
I think this does not help. There probably must be a trick to use along with the fubini's theorem
– Ricardo Freire
Nov 21 '18 at 22:31
Fubini is obtained from approximating with functions in $C^0_c$. What fails when approximating with functions in $C^1$ ?
– reuns
Nov 22 '18 at 18:40
Fubini is obtained from approximating with functions in $C^0_c$. What fails when approximating with functions in $C^1$ ?
– reuns
Nov 22 '18 at 18:40
add a comment |
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Did you try approximating with functions in $C^1$ ?
– reuns
Nov 21 '18 at 21:48
I think this does not help. There probably must be a trick to use along with the fubini's theorem
– Ricardo Freire
Nov 21 '18 at 22:31
Fubini is obtained from approximating with functions in $C^0_c$. What fails when approximating with functions in $C^1$ ?
– reuns
Nov 22 '18 at 18:40