Question

Let ? be the sample space of an experiment and let ℱ be a collection of subsets of ?.

a) What properties must ℱ have if we are to construct a probability measure on (?,ℱ)?

b) Assume ℱ has the properties in part (a). Let ? be a function that maps the elements of ℱ onto ℝ such that

i) ?(?) ≥ 0 , ∀ ? ∈ ℱ ii) ?(?) = 1 and iii) If ?1 , ?2 … are disjoint subsets in ℱ then ?(⋃ ??) = ∞ ?=1 ∑ ?(??) ∞ ?=1 . Show that 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 any two subsets of ? that are elements of ℱ.

i) Show that (? ∩ ?) ∈ ℱ.

ii) Show that (? ∖ ?) ∈ ℱ, where (? ∖ ?) is the set of outcomes that are in ? but not in ?.

iii) Show that (? △ ?) ∈ ℱ, where (? △ ?) is the set of outcomes that are either in ? or in ? but not in both.

iv) Let ?1 , ?2 , ?3 … be elements of ℱ. Show that ⋂ ?? ∞ ?=1 ∈ ℱ

Answer 1