Some authors reserve the name Meas for categories whose objects are measure spaces, and denote the category of measurable spaces as Mble, or other notations.
There is a natural forgetful functor to the category of sets which assigns to each measurable space the underlying set and to each measurable map the underlying function.
The forgetful functor U has both a left adjoint which equips a given set with the discrete sigma-algebra, and a right adjoint which equips a given set with the indiscrete or trivial sigma-algebra.
Moreover, since any function between discrete or between indiscrete spaces is measurable, both of these functors give full embeddings of Set into Meas.
In fact, the forgetful functor U : Meas → Set uniquely lifts both limits and colimits and preserves them as well.