  We further develop the \emph{Multivariate Decomposition Method} (MDM) for the
Lebesgue integration of functions of infinitely many variables
$x_1,x_2,x_3,\ldots$ with respect to a corresponding product of a one
dimensional probability measure.
  Although a number of concepts of infinite-dimensional integrals have been
used in the literature, questions of uniqueness and compatibility have mostly
not been studied. We show that, under appropriate convergence conditions, the
Lebesgue integral equals the `anchored' integral, independently of the anchor.
  The MDM assumes that point values of $f_{\mathfrak{u}}$ are available for
important subsets ${\mathfrak{u}}$, at some known cost. In this paper we
introduce a new setting, in which it is assumed that each $f_{\mathfrak{u}}$
belongs to a normed space $F_{\mathfrak{u}}$, and that bounds
$B_{\mathfrak{u}}$ on $\|f_{\mathfrak{u}}\|_{F_{\mathfrak{u}}}$ are known. This
contrasts with the assumption in many papers that weights
$\gamma_{\mathfrak{u}}$, appearing in the norm of the infinite-dimensional
function space, are somehow known. Often such weights $\gamma_{\mathfrak{u}}$
were determined by minimizing an error bound depending on the
$B_{\mathfrak{u}}$, the $\gamma_{\mathfrak{u}}$ \emph{and} the chosen
algorithm, resulting in weights that depend on the algorithm. In contrast, in
this paper only the bounds $B_{\mathfrak{u}}$ are assumed known.
  We give two examples in which we specialize the MDM: in the first case
$F_{\mathfrak{u}}$ is the $|{\mathfrak{u}}|$-fold tensor product of an anchored
reproducing kernel Hilbert space, and in the second case it is a particular
non-Hilbert space for integration over an unbounded domain.
