  We construct a measure-valued equivalent to the spatial Lambda-Fleming-Viot
process (SLFV) introduced in [Eth08]. In contrast with the construction carried
out in [Eth08], we fix the realization of the sequence of reproduction events
and obtain a quenched evolution of the local genetic diversities. To this end,
we use a particle representation which highlights the role of the genealogies
in the attribution of types (or alleles) to the individuals of the population.
This construction also enables us to clarify the state-space of the SLFV and to
derive several path properties of the measure-valued process as well as of the
labeled trees describing the genealogical relations between a sample of
individuals. We complement it with a look-down construction which provides a
particle system whose empirical distribution at time t, seen as a process in t,
has the law of the quenched SLFV. In all these results, the facts that we work
with a fixed configuration of events and that reproduction occurs only locally
in space introduce serious technical issues that are overcome by controlling
the number of events occurring and of particles present in a given area over
macroscopic time intervals.
