Sigma-algebra on $\Omega = \{1,2,3,4\}$ generated by $\epsilon = \{\{1,2\}, \{2,3\},\{4\}\}$ Let $\Omega = \{1,...,4\}$.
Let $\epsilon = \{\{1,2\}, \{2,3\},\{4\}\}$. Find $\sigma$ ($\epsilon$), generated by ($\epsilon$), and justify answer.
Could someone please give me some direction and help with this question please.
If I say $\{1,2\}=A$, $\{2,3\}=B$ and $\{4\}=C$ then $A^c=\{3,4\}$, $B^c=\{1,4\}$ and $C^c=\{1,2,3\}$.
I know that at least Omega and the empty set has to be part of the Sigma-algebra as well as the single partitions and their complements:
$\sigma(\epsilon)=\{\emptyset, \Omega, A, B, C, A^c, B^c, C^c\}$
Additionally, there has to be the union of the elements inside and this is the point I am not sure about. Are the following elements part of the Sigma-algebra as well (+ what about the complements?)
$$ A \cup B = \{1,2,3\}$$
$$ A \cup C = \{1,2,4\}$$
$$ B \cup C = \{2,3,4\} $$
$$ A \cup B \cup C = \Omega $$
 A: Here is an interesting theorem – it will help you solve more exercises in this vein and it's a hint to prove that σ-algebra cannot be countable.
Definition. An atom of a σ-algebra $Σ$ is a non-void set $Á ∈ Σ$ that contains no other set of $Σ$.
Example. If a singleton belongs to σ-algebra, then it's an atom.
Theorem. If σ-algebra has $N \in \mathbb{N}$ atoms, then it has $2^N$ elements.
Proof. All other elements of σ-algebra are just unions of atoms. So we are actually asking a combinatorial question "In how many ways we can take unions of $N$ sets?" For $0 ≤ k ≤ N$, we have ${N \choose k}$ possible unions of $k$ sets (for $k=0$ we get an empty set, and for $k=N$ the whole space). In total
$$
\sum_{k=0}^N {N \choose k} = \text{number of all subsets of $N$ elements set} = 2^N.
$$
$\Box$
With this theorem, you can be sure that you haven't missed any element of σ-algebra.
Answer. In your exercise surely $\{4\}$ is an atom. But by taking intersections and differences you will discover that $\{1\}, \{2\}, \{3\}$ are also atoms. Thus your σ-algebra will have $2^4$ elements and it has to be a power set.

Bonus. Intuitively σ-algebra generated by a finite family of sets $\mathcal{G}$ consists of sets made by all possible combinations (intersections, sums, complements, ...). Concretely, here are a Venn Diagrams for all possible combinations for two sets:

On the left, you have all possible operations "inside". On the right the complements. This gives you a systematic way to generate all elements of σ-algebra:




"Inside"
Complements




$A$
$A^c$


$B$
$B^c$


$A \cup B$
$(A \cup B)^c$


$A \cap B$
$(A \cap B)^c$


$A \setminus B$
$(A \setminus B)^c$


$B \setminus A$
$(B \setminus A)^c$


$A \triangle B$
$(A \triangle B)^c$




Add an empty set, a whole space, and voilà! But it gets much messier for more than two sets. And that's why you see only exercises with small generators .
