  In this paper we introduce a new class of integralgeometric measures in
$\mathbb{R}^n$, built upon the idea of slicing, and depending on the dimension
$0 \leq m \leq n$ and on the exponent $p \in [1,\infty]$. Among this class we
find general conditions which guarantee rectifiability. Two main consequences
follow. The solution to a long standing open problem proposed by Federer and
concerning the rectifiability of the Integralgeometric measure with exponent $p
\in (1,\infty]$, as well as a novel criterion of rectifiability via slicing for
arbitrary Radon measures. The latter is reminiscent of the rectifiable slices
theorem originally discovered by White for flat chains and by Ambrosio and
Kircheim for metric currents. As far as we know, such a criterion is the first
result that sheds light in the understanding of rectifiability of Radon
measures by slicing. For this reason its proof requires a completely new
technique. Eventually, an alternative proof of the closure theorem for flat
chains with discrete-group coefficients is provided.
