σ-algebra

Dependencies:

1. /sets-and-relations/countable-set

$\newcommand{\Fcal}{\mathcal{F}}$ $\newcommand{\Scal}{\mathcal{S}}$ Let $X$ be a set (called 'ground set') and let $\Fcal$ be a subset of the power-set of $X$. We say that $(X, \Fcal)$ is a $\sigma$-algebra (or $\Fcal$ is a $\sigma$-algebra over $X$) iff all of these properties hold:

1. $X \in \Fcal$.
2. closure under complementation: $A \in \Fcal \implies X-A \in \Fcal$. $X-A$ is called the complement of $A$, and is denoted by $\overline{A}$.
3. closure under countable unions: Let $\Scal = \{A_1, A_2, \ldots\} \subseteq \Fcal$ be a countable set. Then $\left(\bigcup_{A \in \Scal} A\right) \in \Fcal$.

As a trivial example, $\{X, \emptyset\}$ is a $\sigma$-algebra over $X$. The power-set of $X$ is a $\sigma$-algebra over $X$ if $X$ is finite.

Dependency for:

1. Measurable function
2. Generated σ-algebra
3. Measure
4. σ-algebra is closed under countable intersections
5. Set function
6. Probability
7. Independence of random variables (incomplete)
8. X and Y are independent implies X and f(Y) are independent
9. Fair division

Info:

• Depth: 1
• Number of transitive dependencies: 1

Transitive dependencies:

1. /sets-and-relations/countable-set