Let ? be the sample space of an experiment and let ℱ be acollection of subsets of ?.
a) What properties must ℱ have if we are to construct aprobability measure on (?,ℱ)?
b) Assume ℱ has the properties in part (a). Let ? be a functionthat maps the elements of ℱ onto ℠such that
i) ?(?) ≥ 0 , ∀ ? ∈ ℱ ii) ?(?) = 1 and iii) If ?1 , ?2 … aredisjoint subsets in ℱ then ?(⋃ ??) = ∞ ?=1 ∑ ?(??) ∞ ?=1 . Showthat 0 ≤ ?(?) ≤ 1, ∀? ∈ ℱ.
c) Is every subset of ? necessarily an event? Explain briefly.Rigorous definitions are not necessary.
d) Assume ℱ has the properties in part (a). Let ? and ? be anytwo subsets of ? that are elements of ℱ.
i) Show that (? ∩ ?) ∈ ℱ.
ii) Show that (? ∖ ?) ∈ ℱ, where (? ∖ ?) is the set of outcomesthat are in ? but not in ?.
iii) Show that (? △ ?) ∈ ℱ, where (? △ ?) is the set of outcomesthat are either in ? or in ? but not in both.
iv) Let ?1 , ?2 , ?3 … be elements of ℱ. Show that ⋂ ?? ∞ ?=1 ∈ℱ
