  This article is a fundamental study in computable measure theory. We use the
framework of TTE, the representation approach, where computability on an
abstract set X is defined by representing its elements with concrete "names",
possibly countably infinite, over some alphabet {\Sigma}. As a basic
computability structure we consider a computable measure on a computable
$\sigma$-algebra. We introduce and compare w.r.t. reducibility several natural
representations of measurable sets. They are admissible and generally form four
different equivalence classes. We then compare our representations with those
introduced by Y. Wu and D. Ding in 2005 and 2006 and claim that one of our
representations is the most useful one for studying computability on measurable
functions.
