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in the book "Limit theorems of stochastic processes" by Jacod/Shyraev $\nu$ is defined as the set of all real valued prcesses $A$ with $A_0=0$ that are cadlag, adapted and whose each path has finite variation over each finite interval $[0,t]$.

The book now says that each path of an element of $\nu$ can be identified with a signed measure. Can someone provide me with a link to a proof?

I know that there is a bijection between the set of measures and the set of distribution functions on $\mathbb{R}$.

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  • $\begingroup$ $\mu([b,c]) = f(c)-f(b)$ ? $\endgroup$ – reuns Jul 31 '16 at 13:23
  • $\begingroup$ If f is your distribution function, then this defines a probability measure which has f as distribution function. Or what do you mean by your comment? $\endgroup$ – peer Jul 31 '16 at 13:29
  • $\begingroup$ $f$ is a path of one of your process $\endgroup$ – reuns Jul 31 '16 at 13:30
  • $\begingroup$ @user1952009 I see. Do you have a link for the proof? I assume the proof for measures can be used, but to have the proof would be better for me. $\endgroup$ – peer Aug 3 '16 at 15:30
  • $\begingroup$ a proof of what ? it seems obvious to me $\mu$ is a signed measure $\endgroup$ – reuns Aug 3 '16 at 15:39
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Summarizing comments of @user1952009, given a path of your stochastic process $A(\cdot, \omega)$ over $t\in \Bbb R$ you can define a measure $\mu_{A(\cdot, \omega)}$ as follows: $$ \mu_{A(\cdot, \omega)}([s,y]) = A(t,\omega) - A(s,\omega) \tag{1} $$ and extend it to all Borel subsets of $\Bbb R$ using e.g. Caratheodory extension theorem. Since $A$ does not have to be monotonically increasing, in general $\mu_{A(\cdot, \omega)}$ is a signed (rather than positive) measure. One condition that signed measure has to satisfy is to have a finite total variation, which is secured by the fact that each sample path of $A$ is of finite variation.

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