  We elaborate on anomaly induced actions of the Wess-Zumino (WZ) form and
their relation to the renormalized effective action, which is defined by an
ordinary path integral over a conformal sector, in an external gravitational
background. In anomaly-induced actions, the issue of scale breaking is usually
not addressed, since these actions are obtained only by solving the trace
anomaly constraint and are determined by scale invariant functionals. We
investigate the changes induced in the structure of such actions once
identified in dimensional renormalization (DR) when the $\epsilon = d-4 \to 0$
limit is accompanied by the dimensional reduction (DRed) of the field
dependencies. We show that operatorial nonlocal modifications
$(\sim\Box^\epsilon)$ of the counterterms are unnecessary to justify a scale
anomaly. In this case, only the ordinary finite subtractions play a critical
role in the determination of the scale breaking. This is illustrated for the WZ
form of the effective action and its WZ consistency condition, as seen from a
renormalization procedure. Logarithmic corrections from finite subtractions are
also illustrated in a pure $d=4$ (cutoff) scheme. The interplay between two
renormalization schemes, one based on dimensional regularization (DR) and the
second on a cutoff in $d=4$, illustrates the ambiguities of DR in handling the
quantum corrections in a curved background. Therefore, using DR in a curved
background, the scale and trace anomalies can both be obtained by counterterms
that are Weyl invariant only at $d=4$.
