We calculate the reduced density matrix for the inflaton field in a model of chaotic inflation by tracing out degrees of freedom corresponding to various bosonic fields. Apart from the standard decoherence factor there also arises a contribution from the Euclidean effective action of quantum fields. We regularise the ultraviolet divergences in the decoherence factor. Dimensional regularisation does not work because it would lead to a density matrix which does not define a bounded operator. A conformal redefinition of the environmental fields leads to a bounded density matrix. This redefinition is essentially fixed by the requirement that decoherence be absent for vanishing particle creation. Off-diagonal elements of the reduced density matrix become negligibly small for non-conformally invariant fields, so that due to bosonic fields the Universe can acquire classical properties near the onset of inflation.