The paper investigates the completeness of the space of separable measures on a metric space X with respect to the Kantorovich-Rubinshtein (Hutchinson) metric. It proves that the space of separable measures on X is complete if and only if X itself is complete. This result is applied to the theory of self-similar fractals, where it ensures the existence and uniqueness of invariant measures for countable systems of contractions. The paper also addresses the completeness of the space of measures in the Hutchinson metric, showing that it is equivalent to the completeness of the underlying metric space. The key result is that the space of separable measures on X is complete in the Hutchinson metric if and only if X is complete. This extends Hutchinson's theorem to countable systems of contractions and provides a foundation for analyzing invariant measures on non-compact spaces. The paper also includes a detailed analysis of the properties of separable measures, the metrization of the space of measures, and the convergence of sequences of measures in the Hutchinson metric. The results have important implications for the study of fractals and self-similar sets, particularly in the context of infinite iterated function systems.The paper investigates the completeness of the space of separable measures on a metric space X with respect to the Kantorovich-Rubinshtein (Hutchinson) metric. It proves that the space of separable measures on X is complete if and only if X itself is complete. This result is applied to the theory of self-similar fractals, where it ensures the existence and uniqueness of invariant measures for countable systems of contractions. The paper also addresses the completeness of the space of measures in the Hutchinson metric, showing that it is equivalent to the completeness of the underlying metric space. The key result is that the space of separable measures on X is complete in the Hutchinson metric if and only if X is complete. This extends Hutchinson's theorem to countable systems of contractions and provides a foundation for analyzing invariant measures on non-compact spaces. The paper also includes a detailed analysis of the properties of separable measures, the metrization of the space of measures, and the convergence of sequences of measures in the Hutchinson metric. The results have important implications for the study of fractals and self-similar sets, particularly in the context of infinite iterated function systems.