**Definition** A fine measure on $P\_\kappa(\lambda)$ is a non-principal ultrafilter on $P\_\kappa(\lambda)$ which contains all upper cones $\uparrow{x}=\{y\in P\_\kappa(\lambda)|x\subset y\}$, for all $x\in P\_\kappa(\lambda)$.


Bagaria and Magidor defined a cardinal $\kappa$ to be $\aleph\_1$-strongly compact if there exists a non-principal $\aleph\_1$-complete fine measure on $P\_\kappa(\lambda)$, for all $\lambda\ge\kappa$. This is a refinement of the usual definition of strongly compacts where $\kappa$-completeness has been relaxed to $\aleph\_1$-completeness.


I want to further relax the assumption that all upper cones belong to $U$.


**Definition** Let $F$ be a subset of $\lambda$ (which maybe empty).
An ultrafilter on $P\_\kappa(\lambda)$ is $F$-fine if for every $a\in \lambda$, $\uparrow{\{a\}}\in U$ if and only if $a\in F$.
If $F=\lambda$ then we recover the definition of a fine measure.


**Question**
Assume $\kappa$ is an $\aleph\_1$-strongly compact cardinal. Let $F\subset \lambda$ be of size $<\lambda$. Can we prove there is a non-principal $F$-fine $\aleph\_1$-complete ultrafilter on $P\_\kappa(\lambda)$?



$\aleph_1$-complete fine measures on $P_\kappa(\lambda)$
Yes, if $F$ is large enough. Let $j : V \to M$ be the embedding derived from an $\aleph\_1$-strongly compact ultrafilter $U$ on $P\_\kappa(\lambda)$. Let $[\mathrm{id}]$ be the set in $M$ represented by the identity function. Define $W \subseteq P(P\_\kappa(\lambda))$ by:
$$A \in W \Leftrightarrow [\mathrm{id}] \cap j(F) \in j(A).$$


Then $W$ is a countably complete ultrafilter. If $\alpha \in F$, then $j(\alpha) \in [\mathrm{id}] \cap j(F)$, so $[\mathrm{id}] \cap j(F) \in j({\uparrow}\{\alpha\})$. If $\alpha\notin F$, then $j(\alpha) \notin j(F)$, so $[\mathrm{id}] \cap j(F) \notin j({\uparrow}\{\alpha\})$. Thus $W$ is $F$-fine.


It remains to check that $W$ is nonprincipal. This holds when $|F| \geq \kappa$. To see this, suppose on the contrary that there is $z\_0$ such that $A \in W$ iff $z\_0 \in A$. Then $[\mathrm{id}] \cap j(F) = j(z\_0)$. But $|[\mathrm{id}] \cap j(F)| \geq |j[\lambda] \cap j(F)| \geq j(\kappa)$, which means $|z\_0| \geq \kappa$, which is false.


On the other hand, if $|F|<\mathrm{crit}(j)$, then $j(F) = j[F] = [\mathrm{id}] \cap j[F]$, so $A \in W$ iff $j(F) \in j(A)$, thus $A \in W$ iff $F \in A$. In this case, $W$ is principal.


I'm not sure what happens when $\mathrm{crit}(j) \leq |F| < \kappa$.


