For a discrete set X with n elements, there can be finite many Σ = { σ 1 , σ 2 , ⋯ , σ m } algebra defined on it.Given a σ-algebra σ, let S ( σ ) = { A ∈ σ ∣ A ∩ B = ∅ , ∀ B ∈ σ ∖ { A , X } }. And how can I prove that 𝑆 ( σ ) = 𝑆 ( σ ′ ) implies σ = σ ′ ?

Question

For a discrete set   X with   n elements, there can be finite many   Σ  =  {      σ    1    ,      σ    2    ,  ⋯  ,      σ    m    } algebra defined on it.Given a   σ-algebra   σ, let   S  (  σ  )  =  {  A  ∈  σ  ∣  A  ∩  B  =  ∅  ,  ∀  B  ∈  σ  ∖  {  A  ,  X  }  }. And how can I prove that       𝑆    (  σ  )  =      𝑆    (      σ    ′    ) implies   σ  =  σ      ′  ?

Alisa Gilmore · Accepted Answer

Consider   X  =  {      x    1    ,      x    2    ,      x    3    ,      x    4    } and define      σ    1    =  σ  (  {  {      x    1    }  ,  {      x    2    }  }  )    and        σ    2    =  σ  (  {  {      x    3    }  ,  {      x    4    }  }  )  .Then σ1≠σ2 but   S  (      σ    1    )  =  S  (      σ    2    )  =  {  ∅  }.

Zion Wheeler · Accepted Answer

σ    1    =  σ  (  {  {      x    1    }  ,  {      x    2    }  }  ) is the   σ-algebra generated by the sets   {      x    1    } and   {      x    2    }, i.e. the smallest   σ-algebra containing   {      x    1    } and   {      x    2    }.

For a discrete set X with n elements, there can be finite many <mi mathvariant="norm

Answered question

Answer & Explanation

New Questions in Pre-Algebra