  In this paper we start a detailed study of a new notion of rectifiability in
Carnot groups: we say that a Radon measure is $\mathscr{P}_h$-rectifiable, for
$h\in\mathbb N$, if it has positive $h$-lower density and finite $h$-upper
density almost everywhere, and, at almost every point, it admits a unique
tangent measure up to multiples.
  First, we compare $\mathscr{P}_h$-rectifiability with other notions of
rectifiability previously known in the literature in the setting of Carnot
groups, and we prove that it is strictly weaker than them. Second, we prove
several structure properties of $\mathscr{P}_h$-rectifiable measures. Namely,
we prove that the support of a $\mathscr P_h$-rectifiabile measure is almost
everywhere covered by sets satisfying a cone-like property, and in the
particular case of $\mathscr P_h$-rectifiabile measures with complemented
tangents, we show that they are supported on the union of intrinsically
Lipschitz and differentiable graphs. Such a covering property is used to prove
the main result of this paper: we show that a $\mathscr{P}_h$-rectifiable
measure has almost everywhere positive and finite $h$-density whenever the
tangents admit at least one complementary subgroup.
