If $F, F':{\bf R} \rightarrow {\bf R}$ are monotone non-decreasing functions, show that $\mu_F = \mu_{F'}$ if and only if there exists a constant $C \in {\bf R}$ such that $F_+(x) = F'_+(x) + C$ and $F_-(x) = F'_-(x) + C$ for all $x \in {\bf R}$. Note that this implies that the value of $F$ at its points of discontinuity are irrelevant for the purposes of determining the Lebesgue-Stieltjes measure $\mu_F$; in particular, $\mu_F = \mu_{F_+} = \mu_{F_-}$.

Homepage › Solution manuals › Terence Tao › An Introduction to Measure Theory › Exercise 1.7.12 (Near uniqueness)

Exercise 1.7.12 (Near uniqueness)

If $F, F^{'} : R \to R$ are monotone non-decreasing functions, show that $μ_{F} = μ_{F^{'}}$ if and only if there exists a constant $C \in R$ such that $F_{+} (x) = F_{+}^{'} (x) + C$ and $F_{−} (x) = F_{−}^{'} (x) + C$ for all $x \in R$ . Note that this implies that the value of $F$ at its points of discontinuity are irrelevant for the purposes of determining the Lebesgue-Stieltjes measure $μ_{F}$ ; in particular, $μ_{F} = μ_{F_{+}} = μ_{F_{−}}$ .

Exercise 1.7.12 (Near uniqueness)

Add answer