  In this article we develop intersection theory in terms of the
$\mathcal{B}$-group of a reduced analytic space. This group was introduced in a
previous work as an analogue of the Chow group; it is generated by currents
that are direct images of Chern forms and it contains all usual cycles.
However, contrary to Chow classes, the $\mathcal{B}$-classes have well-defined
multiplicities at each point.
  We focus on a $\mathcal{B}$-analogue of the intersection theory based on the
St\"uckrad-Vogel procedure and the join construction in projective space. Our
approach provides global $\mathcal{B}$-classes which satisfy a B\'ezout theorem
and have the expected local intersection numbers. An essential feature is that
we take averages, over various auxiliary choices, by integration. We also
introduce $\mathcal{B}$-analogues of more classical constructions of
intersections using the Gysin map of the diagonal. These constructions are
connected via a $\mathcal{B}$-variant of van Gastel's formulas. Furthermore, we
prove that our intersections coincide with the classical ones on cohomology
level.
