851lecture04

# Theorem 41 consider the uncountable sample space 0 1

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ibilities: (i) if p = 0, then ￿ ￿ p= 0 = 0, r ∈Q1 ,r ￿=1 and (ii) if 0 &lt; p ≤ 1, then r∈Q1 ,r ￿=1 ￿ r∈Q1 ,r ￿=1 p = ∞. In either case, we see that (4.2) cannot be satisﬁed for any choice of p with 0 ≤ p ≤ 1. The conclusion that we are forced to make is that we cannot assign a uniform probability to the set H . That is, H is not an event so P {H } does not exist. We can summarize our work with the following theorem. Theorem 4.1. Consider the uncountable sample space [0, 1] with σ -algebra 2[0,1] , the power set of [0, 1]. There does not exist a probability P : 2[0,1] → [0, 1] satisfying both P {[a, b]} = b − a for all 0 ≤ a ≤ b ≤ 1, and P {A ⊕ r} = P {A} for all A ⊆ [0, 1] and...
View Full Document

## This document was uploaded on 04/07/2014.

Ask a homework question - tutors are online