I read in an old book the following example of a measure: For the set M = Q ∩ [ 0 , 1 ] denote with S the set system of subsets of M of the form Q ∩ I, where I is any interval in [0,1]. Let us define the function μ : S → R as follows: for any set A ∈ S of the form A = Q ∩ I we set μ ( A ) = ℓ ( I ) = b − a ..Then it said without proof that μ is finitely additive, but not σ-additive.As I did not get why I tried to prove it by myself and I tried to show that S is a semi-ring, I guess that is important before I start with the other proof.We have ∅ ∈ Q and furthermore ( Q ∩ I 1 ) ∩ ( Q ∩ I 2 ) = Q ∩ I 1 ∩ I 2 and the union of two closed intervals is either an interval or the disjoint union of two intervals. Then ( Q ∩ I 1 ) ∖ ( Q ∩ I 2 ) is also an interval or the disjoint union of two intervals.Now the proof. I do not quite understand how it cannot be σ-additive. Does it have something in common with Cantor sets? I don´t know how to start the proof here. Any help or explanation (maybe an idea for the beginning of a proof) is appreciated. If there is a proof...

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I read in an old book the following example of a measure: For the set   M  =      Q    ∩  [  0  ,  1  ] denote with   S the set system of subsets of   M of the form       Q    ∩  I, where   I is any interval in [0,1]. Let us define the function   μ  :  S  →      R   as follows: for any set   A  ∈  S of the form   A  =      Q    ∩  I we set   μ  (  A  )  =  ℓ  (  I  )  =  b  −  a  ..Then it said without proof that μ is finitely additive, but not σ-additive.As I did not get why I tried to prove it by myself and I tried to show that   S is a semi-ring, I guess that is important before I start with the other proof.We have   ∅  ∈      Q   and furthermore   (      Q    ∩      I    1    )  ∩  (      Q    ∩      I    2    )  =      Q    ∩      I    1    ∩      I    2   and the union of two closed intervals is either an interval or the disjoint union of two intervals. Then   (      Q    ∩      I    1    )  ∖  (      Q    ∩      I    2    ) is also an interval or the disjoint union of two intervals.Now the proof. I do not quite understand how it cannot be   σ-additive. Does it have something in common with Cantor sets? I don´t know how to start the proof here. Any help or explanation (maybe an idea for the beginning of a proof) is appreciated. If there is a proof...

Danika Rojas · Accepted Answer

If   μ is   σ-additive then for disjoint suitable sets       A    n   it must satisfy:  μ      (          ⋃              n        =        1            ∞              A      n        )    =      ∑          n      =      1              ∞        μ  (      A    n    )From this it can be deduced that for suitable sets       B    n   (not necessarily disjoint) it must satisfy:  μ      (          ⋃              n        =        1            ∞              B      n        )    ≤      ∑          n      =      1              ∞        μ  (      B    n    )Now for every   r  ∈  M choose an interval       I    r   with   r  ∈      S    r    =      Q    ∩      I    r    ∈      S  .If   μ is indeed   σ-additive then:  1  =  μ  (  M  )  =  μ      (          ⋃              r        ∈        M                    S      r        )    ≤      ∑          r      ∈      M        μ  (      S    r    )  =      ∑          r      ∈      M        l  (      I    r    )However it is easy to choose the intervals       I    r   in such a way that:      ∑          r      ∈      M        l  (      I    r    )  &amp;lt;  1and doing so we run into a contradiction.

I read in an old book the following example of a measure: For the set M = <mrow class="MJX

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