State Riemanns Integrability Criterion

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So I'm having a tough time understanding Riemann-Stieltjes and a few of the definitions. I have a question that I think I know the answer to but would like some verification. The question is:



Let f:[a,b]-R be bounded ans $alpha$:[a,b]-R be monotonically increasing.



a) For a partition P of [a,b], define the upper and lower R-S sums with respect to $alpha$.



b)Define what it means for f to be Riemann Stieltjes Integrable with respect to $alpha$



c) State Riemanns Integrability Criterion



My answers
a) U(P,f,$alpha$)=$sumlimits_ i=1^n Mi delta alpha i$ where $delta$i=xi-x(i-1)



L(P,f,$alpha$)=$sumlimits_ i=1^n mi delta alpha i$



b)
infU(P,f,$alpha$)= $int_a^b fdalpha$



$supL(P,f,alpha)=int_a^b fdalpha$



So this is integrable with respect to alpha if these two are equal.



c) For every $epsilon$>0 there exits a partition P of I such that U(P,f,$alpha$)-L(P,f,$alpha$)< $epsilon$



Are these solutions correct?




analysis




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    So I'm having a tough time understanding Riemann-Stieltjes and a few of the definitions. I have a question that I think I know the answer to but would like some verification. The question is:



    Let f:[a,b]-R be bounded ans $alpha$:[a,b]-R be monotonically increasing.



    a) For a partition P of [a,b], define the upper and lower R-S sums with respect to $alpha$.



    b)Define what it means for f to be Riemann Stieltjes Integrable with respect to $alpha$



    c) State Riemanns Integrability Criterion



    My answers
    a) U(P,f,$alpha$)=$sumlimits_ i=1^n Mi delta alpha i$ where $delta$i=xi-x(i-1)



    L(P,f,$alpha$)=$sumlimits_ i=1^n mi delta alpha i$



    b)
    infU(P,f,$alpha$)= $int_a^b fdalpha$



    $supL(P,f,alpha)=int_a^b fdalpha$



    So this is integrable with respect to alpha if these two are equal.



    c) For every $epsilon$>0 there exits a partition P of I such that U(P,f,$alpha$)-L(P,f,$alpha$)< $epsilon$



    Are these solutions correct?




    analysis




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      up vote
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      favorite











      So I'm having a tough time understanding Riemann-Stieltjes and a few of the definitions. I have a question that I think I know the answer to but would like some verification. The question is:



      Let f:[a,b]-R be bounded ans $alpha$:[a,b]-R be monotonically increasing.



      a) For a partition P of [a,b], define the upper and lower R-S sums with respect to $alpha$.



      b)Define what it means for f to be Riemann Stieltjes Integrable with respect to $alpha$



      c) State Riemanns Integrability Criterion



      My answers
      a) U(P,f,$alpha$)=$sumlimits_ i=1^n Mi delta alpha i$ where $delta$i=xi-x(i-1)



      L(P,f,$alpha$)=$sumlimits_ i=1^n mi delta alpha i$



      b)
      infU(P,f,$alpha$)= $int_a^b fdalpha$



      $supL(P,f,alpha)=int_a^b fdalpha$



      So this is integrable with respect to alpha if these two are equal.



      c) For every $epsilon$>0 there exits a partition P of I such that U(P,f,$alpha$)-L(P,f,$alpha$)< $epsilon$



      Are these solutions correct?




      analysis




      share|cite|improve this question











      So I'm having a tough time understanding Riemann-Stieltjes and a few of the definitions. I have a question that I think I know the answer to but would like some verification. The question is:



      Let f:[a,b]-R be bounded ans $alpha$:[a,b]-R be monotonically increasing.



      a) For a partition P of [a,b], define the upper and lower R-S sums with respect to $alpha$.



      b)Define what it means for f to be Riemann Stieltjes Integrable with respect to $alpha$



      c) State Riemanns Integrability Criterion



      My answers
      a) U(P,f,$alpha$)=$sumlimits_ i=1^n Mi delta alpha i$ where $delta$i=xi-x(i-1)



      L(P,f,$alpha$)=$sumlimits_ i=1^n mi delta alpha i$



      b)
      infU(P,f,$alpha$)= $int_a^b fdalpha$



      $supL(P,f,alpha)=int_a^b fdalpha$



      So this is integrable with respect to alpha if these two are equal.



      c) For every $epsilon$>0 there exits a partition P of I such that U(P,f,$alpha$)-L(P,f,$alpha$)< $epsilon$



      Are these solutions correct?





      analysis





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      asked Aug 2 at 9:47









      Robbie Meaney

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