Question: It was noted in Section 1.2.1 that statisticians who follow the deFinetti school do not accept the Axiom of Countable Additivity, instead adhering to the
It was noted in Section 1.2.1 that statisticians who follow the deFinetti school do not accept the Axiom of Countable Additivity, instead adhering to the Axiom of Finite Additivity.
(a) Show that the Axiom of Countable Additivity implies Finite Additivity.
(b) Although, by itself, the Axiom of Finite Additivity does not imply Countable Additivity, suppose we supplement it with the following. Let A1 ⊃ A2 ⊃ . . . ⊃ An ⊃ . . . be an infinite sequence of nested sets whose limit is the empty set, which we denote by An ↓ ϕ. Consider the following:
Axiom of Continuity: If An ↓ ϕ, then P(An) → 0.
Prove that the Axiom of Continuity and the Axiom of Finite Additivity imply Countable Additivity.
(a) Show that the Axiom of Countable Additivity implies Finite Additivity.
(b) Although, by itself, the Axiom of Finite Additivity does not imply Countable Additivity, suppose we supplement it with the following. Let A1 ⊃ A2 ⊃ . . . ⊃ An ⊃ . . . be an infinite sequence of nested sets whose limit is the empty set, which we denote by An ↓ ϕ. Consider the following:
Axiom of Continuity: If An ↓ ϕ, then P(An) → 0.
Prove that the Axiom of Continuity and the Axiom of Finite Additivity imply Countable Additivity.
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