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
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