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In this article, we develop a theory of integration on algebraic quantumgroupoids in the form of regular multiplier Hopf algebroids, and establish themain properties of integrals obtained by Van Daele for algebraic quantum groupsbefore - faithfulness, uniqueness up to scaling, existence of a modular elementand existence of a modular automorphism - for algebraic quantum groupoids underreasonable assumptions. The approach to integration developed in this articleforms the basis for the extension of Pontrjagin duality to algebraic quantumgroupoids, and for the passage from algebraic quantum groupoids tooperator-algebraic completions, which both will be studied in separatearticles.

Integration on algebraic quantum groupoids