A little measure stuff

In case the previous post was too heavy ,this would be a better one to start off :

In a (X,μ) space find a(or all such families) minimal set E with μ(E)>0 ,
that is,any proper  μ-measurable subset of E has μ-measure 0 .

So does there exist any in general?

Can you define such (X,μ ) with the desired property ?

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