  We prove that a variant of the classical Sobolev space of first-order
dominating mixed smoothness is equivalent (under a certain condition) to the
unanchored ANOVA space on $\mathbb{R}^d$, for $d \geq 1$. Both spaces are
Hilbert spaces involving weight functions, which determine the behaviour as
different variables tend to $\pm \infty$, and weight parameters, which
represent the influence of different subsets of variables. The unanchored ANOVA
space on $\mathbb{R}^d$ was initially introduced by Nichols & Kuo in 2014 to
analyse the error of quasi-Monte Carlo (QMC) approximations for integrals on
unbounded domains; whereas the classical Sobolev space of dominating mixed
smoothness was used as the setting in a series of papers by Griebel, Kuo &
Sloan on the smoothing effect of integration, in an effort to develop a
rigorous theory on why QMC methods work so well for certain non-smooth
integrands with kinks or jumps coming from option pricing problems. In this
same setting, Griewank, Kuo, Le\"ovey & Sloan in 2018 subsequently extended
these ideas by developing a practical smoothing by preintegration technique to
approximate integrals of such functions with kinks or jumps.
  We first prove the equivalence in one dimension (itself a non-trivial task),
before following a similar, but more complicated, strategy to prove the
equivalence for general dimensions. As a consequence of this equivalence, we
analyse applying QMC combined with a preintegration step to approximate the
fair price of an Asian option, and prove that the error of such an
approximation using $N$ points converges at a rate close to $1/N$.
