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Additivity of riemann stieltjes integral

  1. Jul 16, 2011 #1
    1. The problem statement, all variables and given/known data

    Hi! i'm studying line integral in vector calculus and i've faced a difficulty related with the additivity of line integral.
    I really hope to get an answer for my question through this site.

    (2.17) Theorem
    : If [itex]\int[/itex]fd[itex]\phi[/itex] from a to b exists and a<c<b, then [itex]\int[/itex]fd[itex]\phi[/itex] from a to c and [itex]\int[/itex]fd[itex]\phi[/itex] from c to b both exist and
    [itex]\int[/itex]fd[itex]\phi[/itex]from a to b = [itex]\int[/itex]fd[itex]\phi[/itex] from a to c + [itex]\int[/itex]fd[itex]\phi[/itex] from c to b.

    I refer to this theorem in the text measure and integral by wheeden and zygmund.

    The book lets me know the fact such that the converse of the theorem is not true if both f and [itex]\phi[/itex] have discontinuity at c.

    What I want to know is to check if following statement is true; [Suppose a<c<b. Assume that not both f and [itex]\phi[/itex] are discontinuous at c. If [itex]\int[/itex]fd[itex]\phi[/itex]from a to b and [itex]\int[/itex]fd[itex]\phi[/itex]from c to b exist, then [itex]\int[/itex]fd[itex]\phi[/itex]from a to b exists and
    [itex]\int[/itex]fd[itex]\phi[/itex] from a to b = [itex]\int[/itex]fd[itex]\phi[/itex]from a to c + [itex]\int[/itex]fd[itex]\phi[/itex]from c to b]

    [itex]\epsilon[/itex]-[itex]\delta[/itex] notations are used for the definition of riemann stieltjes integral here.

    In short, I wanna know whether the above statement is true and where the proof of it is written if it exists.
    2. Relevant equations[/b]



    3. The attempt at a solution

    I've tried to prove it by myself and in my proof, I couldn't find the need for the condition 'not both f and [itex]\phi[/itex] are discontinuous at c.'.

    Thank you for reading my long questions!
    I 'll wait for your reply!
     
    Last edited: Jul 16, 2011
  2. jcsd
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