  We present the first steps of a predicative reconstruction of the
constructive Bishop-Cheng measure theory. Working in a semi-formal elaboration
of Bishop's set theory and invoking the notion of a set-indexed family of
subsets (of a given set), we arrive at notions of a pre-integration space and
of a pre-measure space. We then construct the pre-integration space of simple
functions associated to a pre-measure space and the $L^1$-completion of a
pre-integration space. Unlike the standard presentation of Bishop-Cheng measure
theory, our development is completely predicative and avoids the axiom of
countable choice.
