Definition

Sigma-Algebra

Stochastic Calculus · bm.tex

Let $\Omega$ be a nonempty set. A sigma-algebra (or $\sigma$-algebra) on $\Omega$ is a collection $\mathcal{F} \subseteq 2^\Omega$ of subsets satisfying:

$\Omega \in \mathcal{F}$.
If $A \in \mathcal{F}$, then $A^c \in \mathcal{F}$ (closed under complements).
If $A_1, A_2, \ldots \in \mathcal{F}$, then $\bigcup_{n=1}^\infty A_n \in \mathcal{F}$ (closed under countable unions).

The pair $(\Omega, \mathcal{F})$ is called a measurable space, and elements of $\mathcal{F}$ are called measurable sets or events.

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Dependency Graph

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