Suppose that $F,G in NBV$ and $-infty <a<b< infty$, how show that












0














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?










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  • Did you try approximating with functions in $C^1$ ?
    – reuns
    Nov 21 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 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 at 18:40


















0














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?










share|cite|improve this question






















  • Did you try approximating with functions in $C^1$ ?
    – reuns
    Nov 21 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 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 at 18:40
















0












0








0







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?










share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 21 at 20:19









Ricardo Freire

392110




392110












  • Did you try approximating with functions in $C^1$ ?
    – reuns
    Nov 21 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 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 at 18:40




















  • Did you try approximating with functions in $C^1$ ?
    – reuns
    Nov 21 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 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 at 18:40


















Did you try approximating with functions in $C^1$ ?
– reuns
Nov 21 at 21:48




Did you try approximating with functions in $C^1$ ?
– reuns
Nov 21 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 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 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 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 at 18:40

















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