σ-algebra is closed under countable intersections

Dependencies:

1. σ-algebra
2. /sets-and-relations/countable-set
3. /sets-and-relations/de-morgan-laws

$\newcommand{\Fcal}{\mathcal{F}}$ $\newcommand{\Scal}{\mathcal{S}}$ Let $\Fcal$ be a $\sigma$-algebra over $X$. Let $\Scal = \{A_1, A_2, \ldots\}$ be a countable subset of $\Fcal$. Then \[ T = \bigcap_{A \in \Scal} A \in \Fcal \]

Proof

Let $\Scal' = \{\overline{A}: A \in \Scal\}$. Since $\Fcal$ is closed under complementation, $\Scal' \subseteq \Fcal$.

By closure under countable unions, we get \[ \bigcup_{B \in \Scal'} B = \bigcup_{A \in \Scal} \overline{A} \in \Fcal \] By De-Morgan's laws, we get \[ \overline{T} = \overline{\bigcap_{A \in \Scal} A} = \bigcup_{A \in \Scal} \overline{A} \in \Fcal \] By closure under complementation, we get $T \in \Fcal$.

Dependency for:

1. Probability
2. Conditional probability (incomplete)

Info:

• Depth: 2
• Number of transitive dependencies: 3

Transitive dependencies:

1. /sets-and-relations/countable-set
2. /sets-and-relations/de-morgan-laws
3. σ-algebra