  We relate the injectivity of the canonical map from $C(B_{E'})$ to
$L_p(\mu)$, where $\mu$ is a regular Borel probability measure on the closed
unit ball $B_{E'}$ of the dual $E'$ of a Banach space $E$ endowed with the
weak* topology, to the existence of injective $p$-summing linear
operators/$p$-dominated homogeneous polynomials defined on $E$ having $\mu$ as
a Pietsch measure. As an application we fill the gap in the proofs of some
results of concerning Pietsch-type factorization of dominated polynomials.
