  This paper contributes to foundations of the geometric measure theory in the
infinite dimensional setting of the configuration space over the Euclidean
space $\mathbb R^n$ equipped with the Poisson measure $\pi$. We first provide a
rigorous meaning and construction of the $m$-codimensional Poisson measure --
formally written as "$(\infty-m)$-dimensional Poisson measure" -- on the
configuration space. We then show that our construction is consistent with
potential analysis by establishing the absolute continuity with respect to
Bessel capacities. Secondly, we introduce three different definitions of BV
functions based on the variational, relaxation and the semigroup approaches,
and prove the equivalence of them. Thirdly, we construct perimeter measures and
introduce the notion of the reduced boundary. We then prove that the perimeter
measure can be expressed by the $1$-codimensional Poisson measure restricted on
the reduced boundary, which is a generalisation of De Giorgi's identity to the
configuration space. Finally, we construct the total variation measures for BV
functions, and prove the Gau{\ss}--Green formula.
