I have the statement from the book:
"As nondecreasing functions have left limits, a right continuous nondecreasing process is cadlag. Therefore, it is clear that $W^ +{\subset}W$, where $W^+$ is the set of nondecreasing processes and $W$ is the set of processes of finite variation".
My question: Are all cadlag processes of finite variation?
No. Suppose $\sum a_n$ is a convergent, but not absolutely convergent, series, and let $s_k=\sum_{n=1}^ka_n$ be the sequence of partial sums. Define a function $f$ on $[0,1]$ by $f(x)=s_n$ if $x\in[1-2^{1-n},1-2^{-n})$ and $f(1)=\sum_{n=1}^\infty a_n$. Then $f$ is cadlag - the only point where this is not immediate is $1$, but in fact $\lim_{x\uparrow1}f(x)=f(1)$ by construction. However, $f$ has infinite total variation. To see this, consider the partition $\{x_0,\ldots,x_n,x_{n+1}\}$ where $x_k=1-2^{-k}$ for $0\le k\le n$ and $x_{n+1}=1$. Then
$$TV(f)\ge\sum_{k=1}^{n+1}|f(x_k)-f(x_{k-1})|\ge\sum_{k=1}^n|s_{k+1}-s_k|=\sum_{k=1}^n|a_{k+1}|\to\infty$$
as $n\to\infty$.
