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Also, I am not sure the following claim is true: "which assigns a probability to each element of Ω, with the probabilities summing to 1". It *is* true that every sigma algebra must contain $\Omega$, and typically $P(\Omega)=1$. But P acts on $F$, not $\Omega$, and of course not every atom in $\Omega$ must occur in $F$. Since you preface with the claim that you write this partly as an exercise to check understanding of the underlying ideas, I would kindly suggest considering a read-through of chapter 2 of Pollard's excellent "User's Guide to Measure Theoretic Probability". It might clear up some of these matters
I'm not sure I understand how $\Omega$ represents the set of world histories. If world histories were to live anywhere, they'd live in the sigma algebra — as collections of events, per the definition. If not, and every element of $\Omega$ truly is a world history, then how can $F$ represent "information about which events occur in which possible world-histories", when each $f \in F$ is made up of atoms from $\Omega$, that is, when every element in $F$ is a collection of world histories? One of these definitions ought to be recast, I believe. It might be most sensible to make $\Omega$ the set of all possible events across all possible histories, that way you can largely keep your other definitions as-is