Sigma-Algebra Extended by Single Set
Theorem
Let $\Sigma$ be a $\sigma$-algebra on a set $X$.
Let $S \subseteq X$ be a subset of $X$.
For subsets $T \subseteq X$ of $X$, denote $T^\complement$ for the set difference $X \setminus T$.
Then:
- $\map \sigma {\Sigma \cup \set S} = \set {\paren {E_1 \cap S} \cup \paren {E_2 \cap S^\complement}: E_1, E_2 \in \Sigma}$
where $\sigma$ denotes generated $\sigma$-algebra.
Proof
Define $\Sigma'$ as follows:
- $\Sigma' := \set {\paren {E_1 \cap S} \cup \paren {E_2 \cap S^\complement}: E_1, E_2 \in \Sigma}$
Picking $E_1 = X$ and $E_2 = \O$ (allowed by Sigma-Algebra Contains Empty Set), it follows that $S \in \Sigma'$.
On the other hand, for any $E_1 \in \Sigma$, have by Intersection Distributes over Union and Union with Relative Complement:
- $\paren {E_1 \cap S} \cup \paren {E_1 \cap S^\complement} = E_1 \cap \paren {S \cup S^\complement} = E_1 \cap X = E_1$
Hence $E_1 \in \Sigma'$ for all $E_1$, hence $\Sigma \subseteq \Sigma'$.
Therefore, $\Sigma \cup \set S \subseteq \Sigma'$.
Moreover, from Sigma-Algebra Closed under Union, Sigma-Algebra Closed under Intersection and axiom $(2)$ for a $\sigma$-algebra, it is necessarily the case that:
- $\Sigma' \subseteq \map \sigma {\Sigma \cup \set S}$
It will thence suffice to demonstrate that $\Sigma'$ is a $\sigma$-algebra.
Since $X \in \Sigma$, also $X \in \Sigma'$.
Next, for any $E_1, E_2 \in \Sigma$, observe:
\(\ds \paren {\paren {E_1 \cap S} \cup \paren {E_2 \cap S^\complement} }^\complement\) | \(=\) | \(\ds \paren {E_1 \cap S}^\complement \cap \paren {E_2 \cap S^\complement}^\complement\) | De Morgan's Laws: Difference with Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {E_1^\complement \cup S^\complement} \cap \paren {E_2^\complement \cup S}\) | De Morgan's Laws: Difference with Intersection, Set Difference with Set Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {E_1^\complement \cup S^\complement} \cap E_2^\complement} \cup \paren {\paren {E_1^\complement \cup S^\complement} \cap S}\) | Intersection Distributes over Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {E_1^\complement \cap E_2^\complement} \cup \paren {E_2^\complement \cap S^\complement} \cup \paren {E_1^\complement \cap S} \cup \paren {S^\complement \cap S}\) | Union Distributes over Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {E_1^\complement \cap E_2^\complement} \cap \paren {S^\complement \cup S} } \cup \paren {E_2^\complement \cap S^\complement} \cup \paren {E_1^\complement \cap S}\) | Union with Relative Complement, Set Difference Intersection with Second Set is Empty Set | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {E_1^\complement \cap E_2^\complement \cap S} \cup \paren {E_1^\complement \cap S} \cup \paren {E_1^\complement \cap E_2^\complement \cap S^\complement} \cup \paren {E_2^\complement \cap S^\complement}\) | Intersection Distributes over Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {\paren {E_1^\complement \cap E_2^\complement} \cup E_1^\complement} \cap S} \cup \paren {\paren {\paren {E_1^\complement \cap E_2^\complement} \cup E_2^\complement} \cap S^\complement}\) | Intersection Distributes over Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {E_1^\complement \cap S} \cup \paren {E_2^\complement \cap S^\complement}\) | Intersection is Subset, Union with Superset is Superset |
As $\Sigma$ is a $\sigma$-algebra, $E_1^\complement, E_2^\complement \in \Sigma$ and so indeed:
- $\paren {\paren {E_1 \cap S} \cup \paren {E_2 \cap S^\complement} }^\complement \in \Sigma'$
Finally, let $\sequence {E_{1, n} }_{n \mathop \in \N}$ and $\sequence {E_{2, n} }_{n \mathop \in \N}$ be sequences in $\Sigma$.
Then:
\(\ds \bigcup_{n \mathop \in \N} \paren {E_{1, n} \cap S} \cup \paren {E_{2, n} \cap S^\complement}\) | \(=\) | \(\ds \paren {\bigcup_{n \mathop \in \N} \paren {E_{1, n} \cap S} } \cup \paren {\bigcup_{n \mathop \in \N} \paren {E_{2, n} \cap S^\complement} }\) | Set Union is Self-Distributive/Families of Sets | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {\bigcup_{n \mathop \in \N} E_{1, n} } \cap S} \cup \paren {\paren {\bigcup_{n \mathop \in \N} E_{2, n} } \cap S^\complement}\) | Union Distributes over Intersection |
Since $\ds \bigcup_{n \mathop \in \N} E_{1, n}, \bigcup_{n \mathop \in \N} E_{2, n} \in \Sigma$, it follows that:
- $\ds \bigcup_{n \mathop \in \N} \paren {E_{1, n} \cap S} \cup \paren {E_{2, n} \cap S^\complement} \in \Sigma'$
Hence it is established that $\Sigma'$ is a $\sigma$-algebra.
It follows that:
- $\ds \map \sigma {\Sigma \cup \set S} = \Sigma'$
$\blacksquare$