  This paper deals with the theory of rectifiability in arbitrary Carnot
groups, and in particular with the study of the notion of
$\mathscr{P}$-rectifiable measure.
  First, we show that in arbitrary Carnot groups the natural
\textit{infinitesimal} definition of rectifiabile measure, i.e., the definition
given in terms of the existence of \textit{flat} tangent measures, is
equivalent to the global definition given in terms of coverings with
intrinsically differentiable graphs, i.e., graphs with \textit{flat} Hausdorff
tangents. In general we do not have the latter equivalence if we ask the
covering to be made of intrinsically Lipschitz graphs.
  Second, we show a geometric area formula for the centered Hausdorff measure
restricted to intrinsically differentiable graphs in arbitrary Carnot groups.
The latter formula extends and strengthens other area formulae obtained in the
literature in the context of Carnot groups.
  As an application, our analysis allows us to prove the intrinsic
$C^1$-rectifiability of almost all the preimages of a large class of Lipschitz
functions between Carnot groups. In particular, from the latter result, we
obtain that any geodesic sphere in a Carnot group equipped with an arbitrary
left-invariant homogeneous distance is intrinsic $C^1$-rectifiable.
