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We develop the theory of invariant random fields in vector bundles. Thespectral decomposition of an invariant random field in a homogeneous vectorbundle generated by an induced representation of a compact connected Lie group$G$ is obtained. We discuss an application to the theory of cosmic microwavebackground, where $G=SO(3)$. A theorem about equivalence of two differentgroups of assumptions in cosmological theories is proved.

Invariant random fields in vector bundles and application to cosmology