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**Unformatted text preview: **ibilities: (i) if p = 0, then
p=
0 = 0,
r ∈Q1 ,r =1 and (ii) if 0 < 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...

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