I am now stopped in reading 7.19 Theorem of Walter Rudin's Real and Complex Analysis.
Let $f : [a,b] \rightarrow \mathbb{R}$ be absolutely continous and define $F(x) = \sup\sum_{i=1}^{N}|f(t_{i})-f(t_{i-1})|$ $ (a\leq x\leq b),$ where the supremum is taken over all N and over all choices of $\{t_{i}\}$ such that $a=t_0<t_1<…<t_N=x.$
Then, the author mentions that if $(\alpha,\beta)\subset [a,b],$ then $F(\beta)-F(\alpha) = \sup\sum_{i=1}^{n}|f(t_{i})-f(t_{i-1})|, – (1)$
the supremum being taken over all $\{t_i\}$ that satisfy $\alpha = t_0 <…<t_n = \beta$.
I felt somewhat uncomfortable with the above sentence, and referred to Folland's real analysis for a similar content.
My first question is whether it is tacitly assumed in Rudin's book that $F(b)< \infty$, or otherwise the case $F(\beta)-F(\alpha) = \infty – \infty$ is excluded, so that the relationship $(1)$ is always well-defined. (Moreover, one of the conclusions in 7.19 theorem is that $F$ is absolutely continuous on $[a,b]$, which implies that at least one of the two should be assumed for the result to be established.) However, as seen in the picture, Folland's book allows for $+\infty$ as a possible value of the total variation function. (If so, then the total variation function can take the value $+\infty$ only at b?)
My second question is how we can assume without loss of generality that $a$ is always one of the subdivision points so that the relationship $(1)$ can be deduced. I feel that there is a logical gap.
I am so sorry that my thought on this part is not fully organized, and so the question is logically clumsy. The proof for 7.19 Theorem is super clear except for only (1). Any comment would be enormously appreciated.
Best Answer
I got it. I am truly feeling that I have forgotten many things that I studied a long time ago. An absolutely continuous function is of bounded variation, and then for any function of bounded variation, the additivity of total variation can be seen.(e.g. Royden)