Just refreshing my understanding of measurable cardinals, the first step (more questions may follow, but one step at a time) is to make sure I understand the conditions: one of them is
For a (an uncountable) measurable cardinal κ, there exists a non-trivial, 0-1-valued measure μ on P(κ) such that there exists an λ<κ such that for any sequence [Aα: α<λ ] of disjoint sets Aα whose elements are smaller-than-κ ordinals, μ([itex]\cup[/itex]{ Aα }) = ∑μ(Aα)
Would not this mean that there would be a β<λ such that μ(Aβ) = 1 and [itex]\forall[/itex]γ<λ, (γ≠ β [itex]\Rightarrow[/itex] μ(Aβ) = 0)?
If not, why not?
P.S. except of course for those sequences for which for all α in the set of indices of the sequence, μ(Aα)=0
For a (an uncountable) measurable cardinal κ, there exists a non-trivial, 0-1-valued measure μ on P(κ) such that there exists an λ<κ such that for any sequence [Aα: α<λ ] of disjoint sets Aα whose elements are smaller-than-κ ordinals, μ([itex]\cup[/itex]{ Aα }) = ∑μ(Aα)
Would not this mean that there would be a β<λ such that μ(Aβ) = 1 and [itex]\forall[/itex]γ<λ, (γ≠ β [itex]\Rightarrow[/itex] μ(Aβ) = 0)?
If not, why not?
P.S. except of course for those sequences for which for all α in the set of indices of the sequence, μ(Aα)=0
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