The multivariate decomposition method for infinite-dimensional integration

We develop the generic "Multivariate Decomposition Method" (MDM) for weighted integration of functions of infinitely many variables $(x_1,x_2,x_3,...)$. The method works for functions that admit a decomposition $f=\sum_{\mathfrak{u}} f_{\mathfrak{u}}$, where $\mathfrak{u}$ runs over all finite subsets of positive integers, and for each $\mathfrak{u}=\{i_1,...,i_k\}$ the function $f_{\mathfrak{u}}$ depends only on $x_{i_1},...,x_{i_k}$. Moreover, we assume that $f_{\mathfrak{u}}$ belongs to a normed space $F_{\mathfrak{u}}$, and that a bound on $\|f_{\mathfrak{u}}\|_{F_{\mathfrak{u}}}$ is known. We also specialize MDM to $F_{\mathfrak{u}}$ being the $|\mathfrak{u}|$-tensor product of an anchored reproducing kernel Hilbert space, or a particular non-Hilbert space.