Let $\mu_l$ be Gaussian measure of covariance $(\sqrt{\Delta + m^2})^{-1}$ on the circle of length $l,$ where $m \geq 0.$ We construct a probability measure $ u_l$ on $L_2(\mu_l)$ whose expectations compute correlation functions in free bosonic string theory. We next construct a function $V \in L_p(\otimes_l u_l)$ for all $p > 1$ such that $$Z(\lambda) := \int \otimes_l d u_l \exp( i \lambda V)$$ is a $C^\infty$ function whose power series expansion computes the partition function of an {\em interacting} bosonic string theory. Similar results hold for correlation functions. We conjecture that the terms arising in this power series expansion for $m = 0$ correspond to Polyakov measure on the moduli of curves.