Let n ∈ N, ( X , A , μ ) be a measure space and E 1 , . . . , E n ∈ A. For each j ∈ { 1 , . . . , n } define C j ∶= { x ∈ X | x ∈ E k for exactly j indices , k ∈ { 1. . . , n } } .I want to find a general expression for all the C j since I need to prove that each C j is a measurable set, I made an example and it is as followsWith n = 3 C 1 = ( E 1 ∩ E 2 c ∩ E 3 c ) ∪ ( E 1 c ∩ E 2 ∩ E 3 c ) ∪ ( E 1 c ∩ E 2 c ∩ E 3 ) C 2 = ( E 1 ∩ E 2 ∩ E 3 c ) ∪ ( E 1 ∩ E 2 c ∩ E 3 ) ∪ ( E 1 c ∩ E 2 c ∩ E 3 ) C 3 = ⋂ k = 1 3 E k So for C n we already have a general expression: C n = ⋂ k = 1 n E k for C 1 : C 1 = ⋃ j = 1 n ( E j ∩ ⋂ k ≠ j n E k c ) I was thinking of introducing permutations but actually I don't know if it works, someone can guide me a bit to express the C j sets in a manageable way

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Let   n  ∈  N,   (  X  ,  A  ,  μ  ) be a measure space and       E    1    ,  .  .  .  ,      E    n    ∈  A. For each   j  ∈  {  1  ,  .  .  .  ,  n  } define      C    j    ∶=  {  x  ∈  X      |    x  ∈      E    k          for exactly           j         indices    ,  k  ∈  {  1.  .  .  ,  n  }  }  .I want to find a general expression for all the       C    j   since I need to prove that each       C    j   is a measurable set, I made an example and it is as followsWith   n  =  3      C    1    =  (      E    1    ∩      E    2    c    ∩      E    3    c    )  ∪  (      E    1    c    ∩      E    2    ∩      E    3    c    )  ∪  (      E    1    c    ∩      E    2    c    ∩      E    3    )      C    2    =  (      E    1    ∩      E    2    ∩      E    3    c    )  ∪  (      E    1    ∩      E    2    c    ∩      E    3    )  ∪  (      E    1    c    ∩      E    2    c    ∩      E    3    )      C    3    =      ⋂          k      =      1        3        E    k  So for       C    n   we already have a general expression:      C    n    =      ⋂          k      =      1        n        E    k  for       C    1  :      C    1    =      ⋃          j      =      1        n        (          E      j        ∩          ⋂              k        ≠        j            n              E      k      c        )  I was thinking of introducing permutations but actually I don&#039;t know if it works, someone can guide me a bit to express the       C    j   sets in a manageable way

Punktatsp · Accepted Answer

You&#039;re already on your way there! Here&#039;s a quick hint that should help you wrap things up:There&#039;s no rules over what you can use as an index set for your unions/intersections. I think you&#039;re struggling because you want to write them as       ⋃          k      =      1        n  , but you can index over basically any set you want! Since we want to preserve measurability, we&#039;ll need the set to be countable, but other than that the world is your oyster.So, for instance, there&#039;s nothing stopping you from using the set      I    3    =            {        (  A  ,  B  )  ∣      |    A      |    =  3  ,  A  ∩  B  =  ∅  ,  A  ∪  B  =  {  1  ,  …  ,  n  }  }  .Then       I    3   is finite (do you see why?), so we can write      C    3    =      ⋃          (      A      ,      B      )      ∈              I        3                  (          ⋂              k        ∈        A                    E      k        ∩          ⋂              k        ∈        B                    E      k      c        )    .Do you see why this gives you exactly the       C    3   that you want? Moreover, do you see how to generalize this to get       C    j   for any   j  ≤  n?As an aside, you can do this with permutations as well, and again by indexing over sets more complicated than   {  1  …  n  } you can write that idea quite cleanly too. For instance, consider      C    3    =      ⋃          σ      ∈                        S                n                  (          ⋂              k        =        1            3              E              σ        k              ∩          ⋂              k        =        4            n              E              σ        k            c        )    .(Here             S        n  n is the set of permutations of n elements)Again, do you see why this works? Do you see how to generalize it to get       C    k   for   k  ≠  3?

opepayflarpws · Accepted Answer

You can just write:      C    j    =      ⋃          S      ⊆      {      1      ,      …      ,      n      }      ,              |            S              |            =      j            (          ⋂              i        ∈        S                    E      i        ∩          ⋂              i        ∉        S                    E      i      c        )  These may not be &quot;standard&quot; notations, but all that matters is that all unions/intersections here are finite, so it remains measurable.

Let n ∈ N , ( X , A , μ ) be a

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