Remarks on Pκλ-combinatorics

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