If $f$ and $g$ are both Riemann-Stieltjes Integrable with respect to a monotonic function $\alpha$, is it true that $f(g(x))$ is still integrable with respect to $\alpha$?
Riemann-Stieltjes Integrable
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real-analysis
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riemann-integration
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1 Answers
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No. Take $\alpha(x) = 1/x^3$ and $f(x) = g(x) = x^2$. we have $ \int_1^\infty f(x) d\alpha(x) = \int_1^\infty g(x) d\alpha(x) = -3\int_1^\infty \frac 1 {x^2} dx = -3 $ but $ \int_1^\infty f(g(x)) d\alpha(x) = -3\int_1^\infty dx = -\infty $
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0@Albert the second function you defined$f(x) = x^2$& g(x) = x^-1 might not work out. fog(x) has only one point of discontinuity. any function with finite points of discontinuity in Interval$[a,b]$is Reimann Integrable provided α(x) is continuous at those points. – 2012-12-10