For example， fineness of U already implies that N satisfies a version of covering: If A\subseteqλ and |A|amp;amp;amp;amp;lt;δ， then there is a B\inP_{δ}(λ)\cap N with A\subseteq B. But in fact a significantly stronger version of covering holds. To prove it， we first need to recall a nice result due to Solovay， who used it to show that {\sf SCH} holds above a superpact.
Solovay’s Lemma. Let λamp;amp;amp;amp;gt;δ be regular. Then there is a set X with the property that the function f:a\mapsto\sup(a) is injective on X and， for any normal fine measure U on P_δ(λ)， X\inU.
It follows from Solovay’s lemma that any such U is equivalent to a measure on ordinals.
Proof. Let \vec S=\leftamp;amp;amp;amp;lt; S_\alpha\mid\alphaamp;amp;amp;amp;lt;λ\rightamp;amp;amp;amp;gt; be a partition of S^λ_\omega into stationary sets.
(We could just as well use S^λ_{\le\gamma} for any fixed \gammaamp;amp;amp;amp;lt;δ. Recall that
S^λ_{\le\gamma}=\{\alphaamp;amp;amp;amp;lt;λ\mid{\rm cf}(\alpha)\le\gamma\}
and similarly for S^λ_\gamma=S^λ_{=\gamma} and S^λ_{amp;amp;amp;amp;lt;\gamma}.)
It is a well-known result of Solovay that such partitions exist.
Hugh actually gave a quick sketch of a crazy proof of this fact: Otherwise， attempting to produce such a partition ought to fail， and we can therefore obtain an easily definable λ-plete ultrafilter {\mathcal V} on λ. The definability in fact ensures that {\mathcal V}\in V^λ/{\mathcal V}， contradiction. We will encounter a similar definable splitting argument in the third lecture.
Let X consist of those a\inP_δ(λ) such that， letting \beta=\sup(a)， we have {\rm cf}(\beta)amp;amp;amp;amp;gt;\omega， and
a=\{\alphaamp;amp;amp;amp;lt;\beta\mid S_\alpha\cap\beta is stationary in \beta\}.
Then f is 1-1 on X since， by definition， any a\in X can be reconstructed from \vec S and \sup(a). All that needs arguing is that X\inU for any normal fine measure U on P_δ(λ).(This shows that to define U-measure 1 sets， we only need a partition \vec S of S^λ_\omega into stationary sets.)
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