I am following Bernt Øksendal, Stochastic differential equations, 6.ed. Already in chapter 3, he talks about the probability space $(\Omega, \mathcal{F}, P)$, but as far as I can tell, the set $\Omega$ and the sigma algebra $\mathcal{F}$ has not been defined anywhere. Yet, the author states things such as “note that the filtration generated by the Brownian motion is contained in $\mathcal{F}$” on page 25. So, what is the definition of $\Omega$ and $\mathcal{F}$ in such textbooks? Is the idea that this space is the one obtained from the Kolmogorov extension theorem during the construction of the Brownian motion?
2026-03-25 06:03:19.1774418599
In Itô calculus, what is the probability space $(\Omega, \mathcal{F},P)$?
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