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I'm studying from Bobrowski Functional analysis for probability and stochastic processes (not a university course). I got stuck on one of the exercises, exercise 1.3.4.

Let $f:[a,b]\to \mathbb{R}$ ([a,b] compact) where f is of bounded variation. Consider the function $V(t):=V(a,t)$ to be the variation of $f$ on $[a,t]\subseteq [a,b]$. The claim is now that if $f$ is right-continuous, then so is $V$. I.e $V(t+h)-V(t)\to 0$ as $h\downarrow 0$. Since $V(t+h)=V(a,t+h)=V(a,t)+V(t,t+h)=V(t)+V(t,t+h)$ it suffices to show that $V(t,t+h)\to 0$ as $h\downarrow 0$.

To this end I considered the sums of the form $\sum_{k=1}^{n-1}|f(t_{k+1})-f(t_k)|$ where $t=t_1\leq t_2\leq t_3 \leq t_4 \leq ... \leq t_n = t+h$ is a partition of $[t,t+h]$. Given that we choose $h>0$ small enough I can say that each term (due to RC of f) is say at most $\varepsilon$ large. However, this means that the sum is perhaps of order $n\varepsilon$, so I cannot immediately show that $V(t,t+h)\to 0$ from studying the individual terms. Moreover, if f is say a wienerprocess then it has unbounded variation whilst being continuous, so evidently I need to use the combination of bounded variation + right continuity in order to prove the statement. My idea was then to use a kind of dominated convergence/Weierstrass m-test kind of argument. I tried to rewrite the sum as an integral and then tried to use bounded variation to find a dominating function so that I could take point-wise limits. However I have not been able to set this up in a way that worked.

Is there anyone who has some tips on what else I could try? Similar method/or something entirely different. The reason why Bobrowski develops this is in order to define Stieltjes measures and prove the existence of the total variation of mixed measures, so I cannot use any major theorems from this theory in order to prove my statement since this would make it circular.

Kind regards

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Write $f$ as the difference of two right continuous increasing functions. The Jordan decomposition allows us to write $f=f_1-f_2$ where $f_i$ are increasing. For $i \in \{1,2\}$, let $g_i(x) = \lim_{x\to x+}f_i(x)$. The $g_i$ are increasing and right continuous (see proof in this MSE answer). Since $f$ is right continuous, $f=g_1-g_2$.

Now we can bound the total variation as follows:

$$\sum_{k=1}^{n-1}|f(t_{k+1})-f(t_k)| \\ = \sum_{k=1}^{n-1}|(g_1(t_{k+1})-g_2(t_{k+1}))-(g_1(t_k)-g_2(t_k))| \\\le \sum_{k=1}^{n-1}|g_1(t_{k+1})-g_1(t_k)| + \sum_{k=1}^{n-1}|g_2(t_{k+1})-g_2(t_k)| \\ = (g_1(t+h)-g_1(t)) + (g_2(t+h)-g_2(t)) $$

By right continuity, we can choose $h$ small enough to make each of the terms in the last expression less than $\frac 1 2 \epsilon$ for any $\epsilon > 0$.

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