algebra generated by finite set Is algebra generated by a finite set $A$ same a the $\sigma$-algebra generated by the same set $A$?
For example:
$X=\{1,2,3,4\}$, $A=\{\{1,2\},\{ 2,3\},\{ 4\} \}$, what is the algebra generated by $A$ on $X$?
 A: If $\mathcal A=\{A_1,\dots,A_n\}$ is a finite subset of $\wp(X)$ then sets of the form $E_1\cap\cdots\cap E_n$ where $E_i\in\{A_i,A_i^c\}$ all belong to the algebra generated by $\mathcal A$. The sets are disjoint and cover $X$ and the collection of these sets is finite. You can almost speak of a finite partition of $X$ except that the sets are not necessarily non-empty. Then unions of these sets ("equivalence-classes" of the quasi-partition) are automatically finite unions so they will also belong to the algebra generated by $\mathcal A$. Conversely it can be shown that the collection of these unions is an algebra, and also (since the countable unions are actually finite unions) is a $\sigma$-algebra. So it can be identified as algebra and as $\sigma$-algebra generated by $\mathcal A$. 
Consequently the algebra generated by a finite $\mathcal A\subseteq\wp(X)$ coincides with the $\sigma$-algebra generated by $\mathcal A$.
In your example  the sets $E_1\cap\cdots\cap E_n$ form the collection $\{\varnothing,\{1\},\{2\},\{3\},\{4\}\}$. Any subset of $X$ is a union of these sets so the algebra (and $\sigma$-algebra) generated by it is the powerset of $X$.
