Simple question: Let ( E , E ) be a measurable space and Δ ∉ E. Now let E Δ := E ∪ { Δ } and E Δ denote the smallest σ-algebra on E Δ containing E .Question: Can we given an explicit formula for E Δ ?We've clearly got { Δ } ∈ E Δ , since E , E Δ ∈ E Δ .BTW: Is there a better, established notation for the σ-algebra generated by E on E Δ ? Writing σ ( E ) wouldn't make clear whether we mean the σ-algebra generated on E or a larger space like E Δ .

Question

Simple question: Let   (  E  ,      E    ) be a measurable space and   Δ  ∉  E. Now let       E    Δ    :=  E  ∪  {  Δ  } and             E        Δ   denote the smallest σ-algebra on       E    Δ   containing       E  .Question: Can we given an explicit formula for             E        Δ  ?We&#039;ve clearly got   {  Δ  }  ∈            E        Δ  , since   E  ,      E    Δ    ∈            E        Δ  .BTW: Is there a better, established notation for the   σ-algebra generated by       E   on       E    Δ  ? Writing   σ  (      E    ) wouldn&#039;t make clear whether we mean the   σ-algebra generated on   E or a larger space like       E    Δ  .

Harley Fitzpatrick · Accepted Answer

A notation that makes clear,that what you are lookin at is a   σ-Algebra over       E    Δ   is:   σ  (      E    ∪  {  {  Δ  }  }  ). Because       E    ∪  {  {  Δ  }  } is a family of subsets of       E    Δ   but not   E. It&#039;s also discribes the problem you are lookin at. An explicit formula could be:  σ  (      E    ∪  {  {  Δ  }  }  )  =      E    ∪  {  B  ∪  {  Δ  }  :  B  ∈      E    }Tho it remains to be shown that the right side is a sigma algebra and subset of the left side.

Simple question: Let ( E , <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-calig

Answered question

Answer & Explanation

New Questions in Pre-Algebra