  Given a measure space and a totally ordered ordered collection of measurable
sets, called an ordered core, the notion of a core decreasing function is
introduced and used to define the down space of a Banach function space. This
is done using a variant of the K\"othe dual restricted to core decreasing
functions. To study down spaces, the least core decreasing majorant
construction and the level function construction, already known for functions
on the real line, are extended to this general setting. These are used to give
concrete descriptions of the duals of the down spaces and, in the case of
universally rearrangement invariant (u.r.i.) spaces, of the down spaces
themselves.
  The down spaces of $L^1$ and $L^\infty$ are shown to form an exact Calder\'on
couple with divisibility constant $1$; a complete description of the exact
interpolation spaces for the couple is given in terms of level functions; and
the down spaces of u.r.i. spaces are shown to be precisely those interpolation
spaces that have the Fatou property. The dual couple is also an exact
Calder\'on couple with divisibility constant $1$; a complete description of the
exact interpolation spaces for the couple is given in terms of least core
decreasing majorants; and the duals of down spaces of u.r.i. spaces are shown
to be precisely those interpolation spaces that have the Fatou property.
