  We give a construction of the Floer homology of the pair of {\it non-compact}
Lagrangian submanifolds, which satisfies natural continuity property under the
Hamiltonian isotopy which moves the infinity but leaves the intersection set of
the pair compact. This construction uses the concept of Lagrangian cobordism
and certain singular Lagrangian submanifolds. We apply this construction to
conormal bundles (or varieties) in the cotangent bundle, and relate it to a
conjecture made by MacPherson on the intersection theory of the characteristic
Lagrangian cycles associated to the perverse sheaves constructible to a complex
stratification on the complex algebraic manifold.
