Kanamori (Ultrafilters over Uncountable Cardinals) in his Phd Thesis defines a filter $\mathcal F$ as weakly normal whenever every function $f$ such that $\{\xi<\kappa\mid f(\xi)<\xi\}\in\mathcal F$ then there is some $\eta<\kappa$ such that $X=\{\xi<\kappa \mid f(\xi)<\eta\}$ has positive measure (i.e. $X$ has non-empty intersection with every element of $\mathcal F$).
Then he says that weak normality is equivalent to say that every (filter) extension of $\mathcal F$ is weakly normal. But he states it as obvious. I can't find a justification of this.