Let $ (X,\mathscr M,\mu) $ be a measure space. The completion of $ (X,\mathscr M,\mu) $ is the measure space $ (X,\bar{\mathscr M},\bar\mu) $ where $$ \bar{\mathscr M} = \{{E\cup F : \text{$ E\in \mathscr M $ and $ N\subset F $ for some $ N\in \mathscr M $ such that $ \mu(N) = 0 $}}\} $$ and where $ \bar\mu $ is the unique measure on $ \bar{\mathscr M} $ that extends $ \mu $.
I was asking myself if the completion of a measure space as defined above is in some kind the solution of a universal problem (in the same sense that the tensor product $ M\otimes N $ of two modules is the universal space endowed with a bilinear map $ M\times N\to M\otimes N $).