Factorization of measures and perfection

It is proved that a probability measure $P$ defined on a countably generated measurable space $\left( {Y,\mathcal{C}} \right)$ is perfect iff every probability measure on ${\mathbf{R}} \times Y$ having $P$ as marginal can be factored. This result leads to a generalization of a theorem due to Blackwell and Maitra.