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Reading through my lecture notes, and I'm stuck a bit on this concept.

Let $A$ be a set of subsets of $E$. Define $$ \sigma(A) = \{ A \subseteq E \ : \ A \in F \text{ for all } \sigma\text{-algebras } F \text{ containing }A \} .$$ Then $\sigma(A)$ is a $\sigma$-algebra, which is called the $\sigma$-algebra generated by $A$. It is the smallest $\sigma$-algebra containing $A$.

So let's say: $E = \{1,2,3\}$. All the possible subsets of $E$ will be $ = \{ \{ \emptyset \}, \{1,2,3\}, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\} \}$. So if we choose $A= \{1 \}$, then what would $\sigma(\{1\})$ equals to?

$$ \sigma(\{1\}) = \{ \{1 \} \subseteq E \ \colon \ \{1 \} \in F \text{ for all } \sigma\text{-algebras } F \text{ containing } \{1\} \} .$$

How would I find all the $\sigma$-algebras $F$ containing $\{1\}$? Thanks.

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    The definition you gave is not correct. What you wrote is the definition of $\sigma(F)$ not $\sigma(A)$. The set $F$ must be a subset of powerset $2^E$. E.g., you might ask what is $\sigma(F)$ for $F=\{\{1\}\}$. See also here: http://en.wikipedia.org/wiki/Sigma-algebra#Generated_.CF.83-algebra2011-11-13
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    You use $A$ for subsets of $E$ but also for collections of subsets of $E$. Predictably, chaos ensues, for example in the definition of $\sigma(A)$.2011-11-13
  • 0
    Ok, I might have been wrong and maybe you really want to define a $\sigma$-algebra generated by one subset of $E$. Anyway, you're using the symbol $A$ in the definition of $\sigma(A)$ in two meanings, you should clarify/correct this definition. The definition in the original version of question is: $\sigma(A)$ = {$A\subseteq E$ : $A \in$ $F$ for all $\sigma$-algebras F containing A}.2011-11-13

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