  In this paper we investigate the relationship between a general existence of
transport maps of optimal couplings with absolutely continuous first marginal
and the property of the background measure called essentially non-branching
introduced by Rajala-Sturm (Calc.Var.PDE 2014). In particular, it is shown that
the qualitative non-degenericity condition introduced by Cavalletti-Huesmann
(Ann. Inst. H. Poincar\'e Anal. Non Lin\`eaire 2015) implies that any
essentially non-branching metric measure space has a unique transport maps
whenever initial measure is absolutely continuous. This generalizes a recently
obtained result by Cavalletti-Mondino (Commun. Contemp. Math. 2017) on
essentially non-branching spaces with the measure contraction condition
$\mathsf{MCP}(K,N)$.
  In the end we prove a measure rigidity result showing that any two
essentially non-branching, qualitatively non-degenerate measures on a fixed
metric spaces must be mutually absolutely continuous. This result was obtained
under stronger conditions by Cavalletti-Mondino (Adv.Math. 2016). It applies,
in particular, to metric measure spaces with generalized finite dimensional
Ricci curvature bounded from below.
