  We consider decoupling inequalities for random variables taking values in a
Banach space $X$. We restrict the class of distributions that appear as
conditional distributions while decoupling and show that each adapted process
can be approximated by a Haar type expansion in which only the same conditional
distributions appear. Moreover, we show that in our framework a progressive
enlargement of the underlying filtration does not effect the decoupling
properties (e.g., the constants involved). As special case we deal with
one-sided moment inequalities when decoupling dyadic (i.e., Paley-Walsh)
martingales. We establish the decoupling constant of $\mathbb{R}^d$ with the
$l^{\infty}$-norm. As an example of an application, we demonstrate that
Burkholder-Davis-Gundy type inequalities for stochastic integrals of $X$-valued
processes can be obtained from decoupling inequalities for $X$-valued dyadic
martingales.
