Tannaka–Krein duality for compact groupoids I, Representation theory

In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka–Krein duality for compact groups. In this part we study continuous representations of compact groupoids. We show that irreducible representations have finite dimensional fibres. We prove Schur's lemma and Peter–Weyl theorem for compact groupoids.