This article discusses the completeness of the space of separable measures on a metric space $X$ under the Kantorovich-Rubinshtein metric, also known as the Hutchinson distance. The main result is that the space $M(X)$ of separable measures is complete if and only if the underlying metric space $X$ is complete. The author proves this by analyzing the properties of measures and their convergence under the metric $H$, which is defined using Lipschitz functions. The paper begins by introducing key concepts such as separable measures, the Kantorovich-Rubinshtein metric, and the role of the Hutchinson distance in the study of fractals. It then presents a detailed proof of the completeness of $M(X)$, showing that if $X$ is complete, then any Cauchy sequence of measures in $M(X)$ converges to a measure in $M(X)$. Conversely, if $M(X)$ is complete, then $X$ must also be complete. The article also discusses applications of this result to the theory of self-similar fractals, particularly in the context of invariant measures. It shows that the completeness of $M(X)$ ensures the existence of a unique invariant measure for a countable system of contractions, which is essential for the study of fractal geometry. Additionally, the paper addresses the relationship between the space of separable measures and outer measures, proving that the space of separable outer measures is isometric to $M(X)$. The results are significant in the context of measure theory and fractal geometry, as they provide a rigorous foundation for the study of invariant measures and their properties in general metric spaces. The paper also includes a correction to a previous proof and acknowledges contributions from other researchers.This article discusses the completeness of the space of separable measures on a metric space $X$ under the Kantorovich-Rubinshtein metric, also known as the Hutchinson distance. The main result is that the space $M(X)$ of separable measures is complete if and only if the underlying metric space $X$ is complete. The author proves this by analyzing the properties of measures and their convergence under the metric $H$, which is defined using Lipschitz functions. The paper begins by introducing key concepts such as separable measures, the Kantorovich-Rubinshtein metric, and the role of the Hutchinson distance in the study of fractals. It then presents a detailed proof of the completeness of $M(X)$, showing that if $X$ is complete, then any Cauchy sequence of measures in $M(X)$ converges to a measure in $M(X)$. Conversely, if $M(X)$ is complete, then $X$ must also be complete. The article also discusses applications of this result to the theory of self-similar fractals, particularly in the context of invariant measures. It shows that the completeness of $M(X)$ ensures the existence of a unique invariant measure for a countable system of contractions, which is essential for the study of fractal geometry. Additionally, the paper addresses the relationship between the space of separable measures and outer measures, proving that the space of separable outer measures is isometric to $M(X)$. The results are significant in the context of measure theory and fractal geometry, as they provide a rigorous foundation for the study of invariant measures and their properties in general metric spaces. The paper also includes a correction to a previous proof and acknowledges contributions from other researchers.