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I'm trying to solve the second statement of the following exercise. It is Exercise 2.5 of "All of Nonparametric Statistics, Larry Wasserman".

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My try:

\begin{align} |T(F)-T(G)| &= \left|\int x dF - \int x dG\right| \\ &= \left|\int x d(F-G)\right| \\ &\leq \int |x|d(|F-G|) \\ &\leq M \int d(|F-G|) \\ &\leq M \sup_{x}|F(x)-G(x)| \end{align}

I can not verify the last inequality. Is it true?

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  • $\begingroup$ Reminds me of the upper bound limit of a Riemann-Stieltjes integral on a closed interval. $\endgroup$
    – User1865345
    Commented Jun 30 at 8:12
  • $\begingroup$ How did you write the last expression? The integration is on a bounded set? $\endgroup$
    – User1865345
    Commented Jun 30 at 8:29
  • $\begingroup$ @User1865345 I wrote the last inequality without proof. It may or may not be true. Anyway, I want to solve the second statement. $\endgroup$
    – urikokp
    Commented Jun 30 at 9:08
  • $\begingroup$ @User1865345 You can assume the integral is defined on a bounded set, because $|X| \leq M$ for some $M$. $\endgroup$
    – urikokp
    Commented Jun 30 at 9:09

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