Let ( 1, 2, 3, 4 , let C (, 1, 2 , 1, 2 , 2, 4 , 3, 4 , (, and define on C as follows ( 1, 2 ) ( 1, 3 ) ( 1, 4 ) ( 3, 4 ) 3, (() 6, (() 0 Next, on P((), define the measures 1 and 2 by taking 1( 1 ) 1 ( 4 ) 2 ( 2 ) 2 ( 3 ) 1, 1( 2 ) 1 ( 3 ) 2 ( 1 ) 2 ( 4 ) 2, Then show that (i) C is not a field (ii)...

i C is not a field because eg 1 2 1 3 1 2 3 C ii Clearly A 0 and 0 The only two disjoint sets whose ...

Let ( = {1, 2, 3, 4}, let C = {(, {1, 2},{1, 2},{2, 4},{ 3, 4},

Question:

Let ( = {ω1, ω2, ω3, ω4}, let C = {(, {ω1, ω2},{ω1, ω2},{ω2, ω4},{ ω3, ω4}, (, and define μ on C as follows:
μ({ω1, ω2}) = μ ({ω1, ω3}) = μ ({ω1, ω4}) = μ ({ω3, ω4}) = 3, μ (() = 6, μ (() = 0.
Next, on P((), define the measures μ1 and μ2 by taking
μ1({ω1}) = μ1 ({ω4}) = μ2 ({ω2}) = μ2 ({ω3}) = 1,
μ1({ω2}) = μ1 ({ω3}) = μ2 ({ω1}) = μ2 ({ω4}) = 2,
Then show that
(i) C is not a field.
(ii) μ is a measure on C.
(iii) Both μ1 and μ2 are extensions of μ (from C to P(()).
(iv) Construct the outer measure μ* (as is defined in Definition 5) by means of μ defined on C.
(v) Conclude that μ* ( μ1 ( μ2 (so that, if the class C is not a field, the extension of (even a finite measure μ on C) need not be unique.