We show how to construct measures on Banach manifolds associated
to supersymmetric quantum field theories. We give three concrete
examples of our construction. The first example is a family $\mu_P^{s,t}$
of measures
on a space of functions on the two-torus, parametrized by a
polynomial $P$ (the Wess-Zumino-Landau-Ginzburg model).
The second is a family $\mu_\cG^{s,t}$ of measures on a space $\cG$ of
maps from $\P^1$ to a Lie group
(the Wess-Zumino-Novikov-Witten model). Finally we study
a family $\mu_{M,G}^{s,t}$ of measures on the product of a space of
connections on the
trivial principal bundle with structure group $G$ on a three-dimensional
manifold $M$ with a space of $\fg$-valued three-forms on $M.$
We show that these measures are positive, and that the measures
$\mu_\cG^{s,t}$
are Borel probability measures.
As an application we show that formulas arising from expectations
in the measures $\mu_\cG^{s,1}$ reproduce formulas discovered by Frenkel
and Zh\
u in
the theory
of vertex operator algebras. We conjecture that a similar computation for
the measures $\mu_{M,SU(2)}^{s,t},$ where $M$ is a homology three-sphere,
will yield the Casson invariant of $M.$