Functions of bounded variation on $\Bbb R$

Question

How can one define the total variation of a function of bounded variation on $\Bbb R$?

i.e., how one can evaluate the total variation on infinite intervals?!

Answer 1

Well, I would do something like this. If say $f(x) \in BV[a,b]\quad\forall [a,b]\subset\mathbf{R}$ then you could define its TV as something like $\lim\limits_{r\rightarrow+\infty}V_{-r}^r(f)$ (if it exists), somewhat close to v.p. of an integral, but that's kind of an ad-hoc suggestion.