  Starting solely with a set of possible prices for a traded asset $S$ (in
infinite discrete time) expressed in units of a numeraire, we explain how to
construct a Daniell type of integral representing prices of integrable
functions depending on the asset. Such functions include the values of simple
dynamic portfolios obtained by trading with $S$ and the numeraire. The space of
elementary integrable functions, i.e. the said portfolio values, is not a
vector lattice. It then follows that the integral is not classical, i.e. it is
not associated to a measure. The essential ingredient in constructing the
integral is a weak version of the no-arbitrage condition but here expressed in
terms of properties of the trajectory space. We also discuss the continuity
conditions imposed by Leinert (Archiv der Mathematik, 1982) and K\"onig
(Mathematische Annalen, 1982) in the abstract theory of non-lattice integration
from a financial point of view and establish some connections between these
continuity conditions and the existence of martingale measures
