Direct integrals are a generalization of direct sums. For a compact group every representation is a direct sum of irreducibles. This property fails in general for non-compact groups. The best one can get for general groups is a direct integral decomposition into factor representations. The latter is a notion more general than irreducibility. For nice groups these notions coincide, and then every unitary representation is a direct integral of irreducible representations.
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(2009). Direct Integrals. In: Principles of Harmonic Analysis. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85469-4_8
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DOI: https://doi.org/10.1007/978-0-387-85469-4_8
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