- #1
uman
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Hello all. In my book (Mathematical Analysis by Apostol), the author states and proves the following theorem for real (i.e. f and alpha are functions from [a,b] to R) Riemann-Stieltjes integrals:
Assume c\in(a,b) and \alpha is such that \alpha(a)=\alpha(x) when a\leq x<c, and \alpha(x)=\alpha(b) if b\geq x > c, with \alpha(a), \alpha(c), and \alpha(b) arbitrary. If either f or \alpha is continuous from the right AND either f or \alpha is continuous from the left, \int_a^b f\,d\alpha exists and is equal to f(c)[\alpha(b)-\alpha(a)]. The author later, when discussing complex Riemann-Stieltjes (f and alpha are functions from [a,b] to C), lists the theorems for the real case that are also true in the complex case, but leaves this one out. However, I have gone over the proof, and it seems to work perfectly for the complex case as well. Am I missing some subtle detail in the proof, or did the author simply not decide to mention it? Does anyone know if this is true for complex Riemann-Stieltjes integrals?
Assume c\in(a,b) and \alpha is such that \alpha(a)=\alpha(x) when a\leq x<c, and \alpha(x)=\alpha(b) if b\geq x > c, with \alpha(a), \alpha(c), and \alpha(b) arbitrary. If either f or \alpha is continuous from the right AND either f or \alpha is continuous from the left, \int_a^b f\,d\alpha exists and is equal to f(c)[\alpha(b)-\alpha(a)]. The author later, when discussing complex Riemann-Stieltjes (f and alpha are functions from [a,b] to C), lists the theorems for the real case that are also true in the complex case, but leaves this one out. However, I have gone over the proof, and it seems to work perfectly for the complex case as well. Am I missing some subtle detail in the proof, or did the author simply not decide to mention it? Does anyone know if this is true for complex Riemann-Stieltjes integrals?
