A Measurability question

Ages after ... 04.03.2020 {Here µ(A) is the measure of A provided it exists } Suppose,E is any arbitrary µ−measurable set with µ(E)>0 .Then :
(1)Does there exist a set S(E,µ) ⊂ E such that

 for all E` with S⊆E`⊂ E ,µ(S)=µ(E`)=µ(E)
And 

For all S`⊂S ,µ( S`) < µ(S) ?

(2) If not,can we make it exist by restricting:
   to  only some particular µ−measurable sets and defining the restriction as κ−measurable sets and call the S(E,µ) as S(F,κ) where F is the κ−counterpart of E (some κ−measurable set)
Or,

Changing ′′ For all S`⊂S ,µ( S`) < µ(S)′′ condition to ′′ For all S`⊂S ,µ( S`) = µ(S)−∈ ′′ for some suitable choice of ∈(µ,E) [may depend on both]

(3) Now if we can somehow make it exist, can we conclude anything about that S(F,κ) ,
And in particular,make any quantitative description by considering them being in ℙ(ℝ)
(3) Now if we can somehow make it exist, can we conclude anything about that S(F,κ) ,
And in particular,make any quantitative description by considering them being in ℙ(ℝ) .

(4) Can we ,if there′s no affirmative answer to (1)−(3),reverting back,rather define a measure on ℙ(ℝ) ,with the above−mentioned prop without compromising on the definition of measure.

(5)If still not,turn the questions into that of outer−measure,or to pre−measure .
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