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If $A_1$ and $A_2$ are two collection of subsets in $\Omega$ (Sample Space), I need to prove that $$\sigma(A_1) \subseteq \sigma(A_2).$$ I understand that there exist minimal unique $\sigma$-algebras generated by $A_1$ & $A_2$ respectively. However, I am not sure what needs to be demonstrated mathematically, in order to prove the subset status.

I tried to construct an example for this. Let A1={1,2} , A2={1,2,3} , Ω={1,2,3,4}

Then,

σ(A1)={∅,Ω,{1,2},{3,4}}

σ(A2)={∅,Ω,{1,2,3},{4}}

How can I proceed beyond this. I am confused as how to interpret the subsets as opposed to elements.

Appreciate your comments. Thank you.

  • 1
    Where is the problem from?2012-09-15
  • 0
    Your slashes are the wrong one by the way. You should use the backslash.2012-09-15
  • 6
    Probably add the hypothesis $A_1\subset A_2$. After this is done, remind us how $\sigma(A)$ was introduced to you.2012-09-15
  • 0
    If by "with the equality" you mean that $\sigma(A_1)=\sigma(A_2)$ might hold, you might want to use $\subseteq$ (produced by `\subseteq`) instead of $\subset$ (produced by `\subset`). Also note that you can get subscripts in $A_1$ and $A_2$ like this: `A_1`, `A_2`.2012-09-15
  • 1
    What did means is that you need some condition in order to say something.2012-09-15
  • 0
    Please correct the question. Do not leave us to guess what you meant to ask.2012-09-15
  • 0
    Your question does not make sense. If $\sigma(A_1) \subseteq \sigma(A_2)$ is true, then the opposite inclusion is also true(because you don't make any assumption on $A_1$ and $A_2$ other than they are collections of subsets of $\Omega$). So the both are equal. This is clearly not true in general. Your example does not make sense, either. $A_1$ and $A_2$ should be collections of subsets of $\Omega$.2012-09-15

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