Any measure absolutely continuous with respect to time-Lebesgue times jump intensity has a diffuse time-coordinate marginal

Let ν be an s-finite Borel measure on jump sizes, and let η be any measure on real time×real jump-size space which is absolutely continuous with respect to Lebesgue-time × ν. The time-coordinate marginal of η is atomless: every singleton time has probability or measure zero. A fixed-time slice {t}×ℝ has product intensity zero since Lebesgue gives no mass to a singleton. Absolute continuity transfers this nullness to η; projecting to time identifies its singleton preimage with the slice. This is the key diffuse-time measure-theoretic prerequisite for canonical Poisson point-family arrival times under the normalised restriction of the product intensity to nonzero-mass partition pieces. For zero-mass pieces the canonical model may use a dummy Dirac point and they must instead be shown inactive almost surely.