A nonperturbative approach to Polyakov string theory in variable dimensions will be developed. The result leads to a reinterpretation of the critical dimension and an effective compactification of the theory. At each stage the variable dimension theory is based on Gaussian measures, so a discussion of Gaussian measures in linear topological vector spaces is included, detailing the sets of measure zero that arise when the inner product is degenerate. The dimension can be taken to ∞ and the limiting sum exists. The limit represents a correction to [Det Δ]−1/2, which arises as a naive limit. The analysis involves an interplay between zero modes and sets of measure zero with respect to the Gaussian measures. Target space isometries require the factoring out of equivalent configurations from the path integral, which necessitates a discussion of functional Haar measures. The inclusion of source terms is considered.
