On Pietsch measures for summing operators and dominated polynomials

We relate the injectivity of the canonical map from $C(B_{E^{\mathrm{\prime }}})$ to $L_p(\mu )$, where $\mu$ is a regular Borel probability measure on the closed unit ball $B_{E^{\mathrm{\prime }}}$ of the dual $E^{\mathrm{\prime }}$ 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.