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Let S be the sample space of an experiment and let ℱ be a non-empty collection...

Let S be the sample space of an experiment and let ℱ be a non-empty collection of subsets of S such that i) ? ∈ ℱ ⇒ ? ′ ∈ ℱ and ii) ?1 ∈ ℱ and ?2 ∈ ℱ ⇒ ?1 ∪ ?2 ∈ ℱ

a) Show that if ?1 ∈ ℱ and ?2 ∈ ℱ then ?1 ∩ ?2 ∈ ℱ .

b) Show that ? ∈ ℱ.

c) Is ℱ necessarily a ?-algebra? Explain briefly. A rigorous proof is not necessary.

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