Scott: GCH on a measure-one set below a measurable κ gives 2^κ = κ⁺

Let $\kappa$ be an uncountable cardinal and let $D$ be a normal measure on $\kappa$ (a nonprincipal, $\kappa$-complete, normal ultrafilter concentrating on the ordinals below $\kappa$; so $\kappa$ is measurable). Suppose GCH holds below $\kappa$ on a set of measure one:

Then

Contrapositively, violating GCH at a measurable cardinal requires violating it on a measure-one set below; this is the link between $\neg$SCH and GCH failures at measurable cardinals described in the source.

Formalization Note For an ordinal $\alpha$, $|\alpha|$ is its cardinality (α.card); the set of cardinals below $\kappa$ has measure one for a normal measure, so this agrees with the source's set of cardinals $\alpha<\kappa$.