The space of probabilistic $1$-Lipschitz maps

Mohammed Bachir (1)

1 Equations d'evolution

We introduce and study the natural notion of probabilistic $1$-Lipschitz maps. We use the space of all probabilistic $1$-Lipschitz maps to give a new method for the construction of probabilistic metric completion (respectively of probabilistic invariant metric group completion). Our construction is of independent interest. We prove that the space of all probabilistic $1$-Lipschitz maps defined on a probabilistic invariant metric group can be endowed with a monoid structure. Next, we explecite the set of all invertible element of this monoid and characterize the probabilistic invariant complete Menger groups by the space of all probabilistic $1$-Lipschitz maps in the sprit of the classical Banach-Stone theorem.

Domains: Functional Analysis [math.FA]; Probability [math.PR]

Mohammed Bachir. The space of probabilistic $1$-Lipschitz maps. Aequationes Mathematicae, 2019, 93 (5), pp.955-983. ⟨hal-01973212⟩