Generated σ-algebra

Dependencies:

1. σ-algebra

$\newcommand{\Fcal}{\mathcal{F}}$ Let $\Fcal$ be a subset of the power-set of $X$.

Define $\sigma(\Fcal)$ as the set containing all the subsets of $X$ that can be made from elements of $\Fcal$ by a countable number of complement, union and intersection operations. Then $\sigma(\Fcal)$ is a $\sigma$-algebra.

The proof follows almost by definition.

$\sigma(\Fcal)$ is called the '$\sigma$-algebra generated by $\Fcal$'.

Dependency for:

1. Random variable
2. Independence of random variables (incomplete)
3. Borel algebra

Info:

• Depth: 2
• Number of transitive dependencies: 2

Transitive dependencies:

1. /sets-and-relations/countable-set
2. σ-algebra