  In this paper we investigate the foundations for analysis in infinitely-many
(independent) variables. We give a topological approach to the construction of
the regular $\s$-finite Kirtadze-Pantsulaia measure on $\R^\iy$ (the usual
completion of the Yamasaki-Kharazishvili measure), which is an infinite
dimensional version of the classical method of constructing Lebesgue measure on
$\R^n$ (see \cite{YA1}, \cite{KH} and \cite{KP2}). First we show that von
Neumann's theory of infinite tensor product Hilbert spaces already implies that
a natural version of Lebesgue measure must exist on $\R^{\iy}$. Using this
insight, we define the canonical version of $L^2[\R^{\iy}, \la_{\iy}]$, which
allows us to construct Lebesgue measure on $\R^{\iy}$ and analogues of Lebesgue
and Gaussian measure for every separable Banach space with a Schauder basis.
When $\mcH$ is a Hilbert space and $\la_{\mcH}$ is Lebesgue measure restricted
to $\mcH$, we define sums and products of unbounded operators and the Gaussian
density for $L^2[\mcH, \la_{\mcH}]$. We show that the Fourier transform induces
two different versions of the Pontryagin duality theory. An interesting new
result is that the character group changes on infinite dimensional spaces when
the Fourier transform is treated as an operator. Since our construction
provides a complete $\s$-finite measure space, the abstract version of Fubini's
theorem allows us to extend Young's inequality to every separable Banach space
with a Schauder basis. We also give constructive examples of partial
differential operators in infinitely many variables and briefly discuss the
famous partial differential equation derived by Phillip Duncan Thompson
\cite{PDT}, on infinite-dimensional phase space to represent an ensemble of
randomly forced two-dimensional viscous flows.
