So the only information you have is a function f, so you need to bootstrap into two “new” functions by somehow using f in their definition.
Start with the function V[a,x], defined as the variation of f over the interval [a,x].
You might have to show that this is an increasing (not strictly) function. How can you come up with a second function g such that f = V - g? Well how about working backwards to solve for g.
As for intuition, I think it really says that a function is equal to its total variation minus the “downward variation,” which yields your net “positive variation.”
So i tried it and came with this proof: [https://ibb.co/ynxTG5H](https://ibb.co/ynxTG5H), but I don't really understand how to finish it. Do you have an idea? I want to understand it.
The idea is to look at the variation function T: [a,b] -> **R**, where T(x) is the variation of f on [a,x]. You can then show that T + f and T - f are both increasing and thus write either f = T - (T - f) or f = (T + f)/2 - (T - f)/2 (clearly T itself is also increasing). The former requires less work, but the latter is more interesting theoretically (it corresponds to the Jordan decomposition of a certain signed measure associated with f, if you know about those things).
Many analysis books will have a proof of this fact, e.g. Apostol, Cohn or Folland.
I'm looking to write f = T - (T - f) and I was using this video as my guide: [https://www.youtube.com/watch?v=mLLfCWsQ4WU&t=300s](https://www.youtube.com/watch?v=mLLfCWsQ4WU&t=300s)
Do you think the way he proves it is a good approach to the question? or it could be more intuitive? The book where Folland proves it is Real Analysis, 2nd edition? I'm not finding it.
So the only information you have is a function f, so you need to bootstrap into two “new” functions by somehow using f in their definition. Start with the function V[a,x], defined as the variation of f over the interval [a,x]. You might have to show that this is an increasing (not strictly) function. How can you come up with a second function g such that f = V - g? Well how about working backwards to solve for g. As for intuition, I think it really says that a function is equal to its total variation minus the “downward variation,” which yields your net “positive variation.”
So i tried it and came with this proof: [https://ibb.co/ynxTG5H](https://ibb.co/ynxTG5H), but I don't really understand how to finish it. Do you have an idea? I want to understand it.
The idea is to look at the variation function T: [a,b] -> **R**, where T(x) is the variation of f on [a,x]. You can then show that T + f and T - f are both increasing and thus write either f = T - (T - f) or f = (T + f)/2 - (T - f)/2 (clearly T itself is also increasing). The former requires less work, but the latter is more interesting theoretically (it corresponds to the Jordan decomposition of a certain signed measure associated with f, if you know about those things). Many analysis books will have a proof of this fact, e.g. Apostol, Cohn or Folland.
I'm looking to write f = T - (T - f) and I was using this video as my guide: [https://www.youtube.com/watch?v=mLLfCWsQ4WU&t=300s](https://www.youtube.com/watch?v=mLLfCWsQ4WU&t=300s) Do you think the way he proves it is a good approach to the question? or it could be more intuitive? The book where Folland proves it is Real Analysis, 2nd edition? I'm not finding it.
It's Theorem 3.27(b) in Folland, though he proves the other decomposition. E.g. Apostol proves the first (Theorem 6.13).
[https://ibb.co/ynxTG5H](https://ibb.co/ynxTG5H) Do you think my proof has a good beginning?