A definition of the Feynman path integral which does not rest on a limiting procedure based on time-slicing has been given by DeWitt-Morette. We present in this paper a discussion of real Gaussian measures and formulate expressions for the quantum statistical partition function directly in terms of measures of integration on the topological vector space ø0 of continuous functions defined on the time intervalT = (t a ,t b ), such thatx(t a ,t b )=0 for allx ø0. We give a definition of a measure for the space ø0 equivalent to the path integral based on the Uhlenbeck-Ornstein probability distribution. We give expressions for the partition function using the Wiener-Feynman measure and the Uhlenbeck-Ornstein measure. As an exercise in the use of the new techniques, we present calculations of moments of potential functions. The techniques will enable one to solve in a rigorous manner practical problems in quantum statistical mechanics.