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In this paper the problem of the existence of an inverse (or projective) limit measure ${\mu }^{\text{'}}$ of an inverse system of measure spaces $(X_i,{\mu }_i)$ is approached by obtaining first a measure $\tilde{\mu }$ on the whole product space ${\prod }_{i\in I}{X}_i$. The measure $\tilde{\mu }$ will have many of the properties of a limit measure provided only that the measures ${\mu }_i$ possess mild regularity properties. It is shown that ${\mu }^{\text{'}}$ can only exist when $\tilde{\mu }$ is itself a “limit” measure in a more general sense, and that ${\mu }^{\text{'}}$ must then be the restriction of...