  The classical integral localization formula for equivariantly closed forms
(Theorem 7.11 in [BGV]) is well-known and requires the acting Lie group to be
compact. It is restated here as Theorem 2. In this article we extend this
result to NONcompact groups. The main result is Theorem 20. Then, using this
generalization, we prove an analogue of the Gauss-Bonnet theorem for
constructible sheaves (Theorem 43). These results can be used to obtain a
generalization of the Riemann-Roch-Hirzebruch integral formula.
