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Abstract

We prove that $\{x \in \mathcal P_\kappa \lambda\mid x \cap \kappa$ is almost $x$-ineffable$\}$ has $p_*({\rm NIn}_{\kappa, \lambda^{<\kappa}})$-measure 1 and $\{x \in\mathcal P_\kappa\lambda\mid x \cap \kappa$ is $x$-ineffable$\}$ has I-measure 1, where $\mathcal I$ is the complete ineffable ideal on $\mathcal P_\kappa\lambda$. As corollaries, we show that $\lambda$-ineffability does not imply complete $\lambda$-ineffability and that almost $\lambda$-ineffability does not imply $\lambda$-ineffability.