I have an exercise in my measure and integration theory course and I'm a little bit stuck with the σ-rings and σ-algebras. The exercise goes as follows:Let X be a set and ε = { { x } | x ∈ X }. Determine σ ( ε ), the smallest σ-algebra that contains epsilon.I know that ε = { { x } | x ∈ X } is the set that contains the singeltons as elements.My first problem when defining the σ-algebra is whether it has to contain the empty set AND the set X, OR the empty set and the set ε? Because this is one of three conditions for forming a σ-algebra.The second condition, that σ ( ε ) has to contain the complement of a subset A ∈ σ ( ε ), is (I think) clear for me. But then the third condition, that it has to contain the the union of all subsets of σ ( ε ), is also unclear. I am not sure how to "build" this union.

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I have an exercise in my measure and integration theory course and I&#039;m a little bit stuck with the   σ-rings and   σ-algebras. The exercise goes as follows:Let   X be a set and   ε  =  {  {  x  }        |      x  ∈  X  }. Determine   σ  (  ε  ), the smallest   σ-algebra that contains epsilon.I know that   ε  =  {  {  x  }        |      x  ∈  X  } is the set that contains the singeltons as elements.My first problem when defining the   σ-algebra is whether it has to contain the empty set AND the set   X, OR the empty set and the set   ε? Because this is one of three conditions for forming a   σ-algebra.The second condition, that   σ  (  ε  ) has to contain the complement of a subset   A  ∈  σ  (  ε  ), is (I think) clear for me. But then the third condition, that it has to contain the the union of all subsets of   σ  (  ε  ), is also unclear. I am not sure how to &quot;build&quot; this union.

bosco84jsrztu · Accepted Answer

Clearly,   σ  (  ϵ  ) must contain all countable sets. For if we have a set   {      a    i    ∣  i  ∈  S  } where   S  ⊆      N  , then define   {      b    i        }          i      ∈              N             by       b    i   to be   {      a    i    } if   i  ∈  S,   ∅ otherwise. Then   {      a    i    ∣  i  ∈  S  }  =      ⋃          i      ∈              N                  b    i  .Thus, we see that   σ  (  ϵ  ) must also contain all sets whose complement is countable. For if   A has a countable complement, then       A    c    ∈  σ  (  ϵ  ), and thus   A  =  (      A    c        )    c    ∈  σ  (  ϵ  ).The question is: is this all of the elements of   σ  (  ϵ  )?In order to show the answer is &quot;yes&quot;, you must demonstrate that   {  A  ⊆  X  ∣  A is countable or       A    c   is countable} is a   σ-algebra. I&#039;ll leave that as an exercise.

I have an exercise in my measure and integration theory course and I'm a little bit stuck with the

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