Why can not we define Lebesgue Stieltjes integral $\int_0^T f dg$ when $g$ is not a bounded variation function on $[0,T]$ ?
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I just learnt from the book " Stochastic Integration and Differential Equations" by Philip E. Protter that if such an integral has to be defined for all continuous $f$ then $g$ is necessarily of bounded variation. The proof uses Banach Steinhauss Theorem. – chandu1729 Mar 28 '13 at 05:05