I'm trying to do exercise $10.7$ of Jech's Set Theory:
If $\kappa$ is a measurable cardinal, then there exists a normal measure on $\mathcal P_{\kappa}(\kappa)$.
For a set $A$, with $|A|\geq \kappa$, a normal measure on $\mathcal P_{\kappa}(A)$ is a $\kappa$-complete ultrafilter $U$ on $\mathcal P_{\kappa}(A)$ which extends the filter generated by the sets $\hat P=\{Q\in \mathcal P_{\kappa}(A):P\subset Q\}$, and such that for any $f:\mathcal P_{\kappa}(A)\rightarrow A$ with $f(P)\in P$ for all $P$ in a set in $U$, $f$ is constant in some element of $U$.
This is what I've tried so far:
As $\kappa$ is inaccessible it is easy to show that $|\mathcal P_{\kappa}(\kappa)|=\kappa$, then fix a bijection $f:\kappa\rightarrow \mathcal P_{\kappa}(\kappa)$. Let $V$ be a nonprincipal $\kappa$-complete ultrafilter on $\kappa$, then $U=f(V)$ is a $\kappa$-complete ultrafilter on $\mathcal P_{\kappa}(\kappa)$. Using the ultrafilter $U$ we can get an ultrafilter on $\mathcal P_{\kappa}(\kappa)$ that is normal:
Define $\equiv$ on $\mathcal P_{\kappa}(\kappa)^{\mathcal P_{\kappa}(\kappa)}$ by $f\equiv g$ iff $\{A\in \mathcal P_{\kappa}(\kappa):f(A)=g(A)\}\in U$, and define $<$ on $\mathcal P_{\kappa}(\kappa)^{\mathcal P_{\kappa}(\kappa)}/\equiv$ by $[f]<[g]$ iff $\{A: f(A)\subsetneq g(A)\}\in U$, one can show $<$ is a well-ordering; as $\in$ is well-founded on $\kappa$. Let $f$ be the least function such that for any $A\in \mathcal P_{\kappa}(\kappa)$ we have $\{B:f(B)\neq A\}\in U$; since $U$ is not principal such functions exist for instance $d(B)=B$. Then using the minimality of $f$ one can show that $f_{-1}(U)$ is a normal ultrafilter on $\mathcal P_{\kappa}(\kappa)$.
The problem is, how to get the normal ultrafilter and make it contain all the sets $\hat P$?, perhaps this can be done using the well-order defined above. I would like to see another approaches to the problem too.
Thanks
Let $j: V \rightarrow M$ be the ultrapower embedding with $\text{crit}(j) = \kappa$. Say $F \subseteq \mathcal{P}_{\kappa}{(\kappa)}$ has measure one if $\kappa \in j(F)$. Now check.
A model-theory free argument: Let $m$ be a normal measure over $\kappa$. Extend $m$ to all of $\mathcal{P}_{\kappa}(\kappa)$, by declaring the set of non ordinals to be null. Now check again.