  In a previous paper, we proved that for any submartingale $(X_t)_{t \geq 0}$
of class $(\Sigma)$, defined on a filtered probability space $(\Omega,
\mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some
technical conditions, one can construct a $\sigma$-finite measure $\mathcal{Q}$
on $(\Omega, \mathcal{F})$, such that for all $t \geq 0$, and for all events
$\Lambda_t \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] =
\mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t]$$ where $g$ is the last
hitting time of zero of the process $X$. Some particular cases of this
construction are related with Brownian penalisation or mathematical finance. In
this note, we give a simpler construction of $\mathcal{Q}$, and we show that an
analog of this measure can also be defined for discrete-time submartingales.
