I don't think it's correct to say "event space" and "sample space" are synonymous — the sample space contains outcomes, whereas the event space would (I assume, although this is never explicitly stated) contain events. Remember that events are "things we might want to consider". An event space, F, is an algebra over the sample space, Ω.
Look at the example of the last match of the 2015 Six Nations, and adopt the notation X_n to represent team X winning by n points, and similarly X_[n,m] to team X winning by between n and m points. Then the sample space here is Ω={..., F_2, F_1, draw, E_1, E_2, ...} , but we don't care about most of these as separate things, since the table ends up the same afterwards. A few possible event spaces, F, are:
F_1 = power set of {E_[26,∞] , notE_[26,∞] } = {∅ , {E_[26,∞]} , {notE_[26,∞]} , Ω} , corresponding to "who wins the Championship", or
F_2 = power set of { F_[8,∞] , F_[1,7] , draw , E_[1,16] , E_[17,25] , E_[26,∞] } , corresponding to the possible placements of the final table.
If you're still not convinced, have a look at the bottom of P.10 of her notes — the probability function maps from such an F to [0,1], so by your definition, you'd be unable to give the event "England win the Six Nations" a probability (however small!).
