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Abstract: In this paper we introduce a framework for harmonic analysis associated to infinite dimensional Riemannian symmetric spaces of classical type, involving the notion of a rigged manifold. We prove the existence (and in some cases uniqueness) of invariant measures associated to riggings of infinite dimensional classical symmetric spaces. The associated L^2 representations decompose uniquely as direct integrals of irreducible spherical representations [The Plancherel formula has been found by Borodin and Olshanskii].