Sigma-algebra ($\mathcal{F}$)

A sigma-algebra on a set $\Omega$ is a collection $\mathcal{F}$ of subsets such that $\Omega \in \mathcal{F}$, $A\in \mathcal{F}$ implies $\Omega \setminus A\in \mathcal{F}$, and $A_n\in \mathcal{F}$ for all $n$ implies ${\bigcup }_n{A}_n\in \mathcal{F}$. Thus it contains the empty set and is closed under countable intersections as well. This is precisely the closure needed to apply countable additivity without leaving the domain of a measure.

For $\Omega =\{a,b,c\}$,is a sigma-algebra. It describes observations that distinguish $a$ from the pair $\{b,c\}$, but cannot distinguish $b$ from $c$. The power set provides a finer sigma-algebra with all eight subsets. Both are valid; choosing a sigma-algebra records which questions the model can answer.

On the real numbers, the Borel sigma-algebra contains all open sets and everything generated from them by these operations. The Lebesgue measurable sets form a larger sigma-algebra, including all subsets of Borel null sets. Closure under countable unions accommodates constructions involving an entire sequence of observations or approximations.

The sigma-algebra generated by a collection is the smallest sigma-algebra containing all sets in that collection. The displayed intersection is over a nonempty family, since the power set is one candidate. An intersection of sigma-algebras contains the whole space and is closed under complements and countable unions, proving it is a sigma-algebra and establishing minimality.

For example, the half-lines $(-\infty ,x]$ generate the Borel sigma-algebra on the real line. They are Borel, and conversely $(-\infty ,b)={\bigcup }_n(-\infty ,b-1/n]$ and $(a,b)=(-\infty ,b)\setminus (-\infty ,a]$ recover every open interval. Preimages of these half-lines similarly generate the sigma-algebra generated by a random variable.

For a measurable space $(\Omega ,\mathcal{F})$ and any subset $E\subseteq \Omega$, the trace sigma-algebra is the displayed family of subsets of $E$. It contains $E=E\cap \Omega$, is closed under relative complements since $E\setminus (E\cap A)=E\cap (\Omega \setminus A)$, and is closed under countable unions. If $E\in \mathcal{F}$, it equals $\{B\in \mathcal{F}:B\subseteq E\}$. Without that measurability assumption, the traces need not be measurable sets in the original sigma-algebra.

A sigma-algebra $\mathcal{G}$ on the same underlying set as $\mathcal{F}$, with $\mathcal{G}\subseteq \mathcal{F}$. In probability theory it represents information retained from the original probability space and is a possible conditioning collection for a conditional expectation.

For a real random variable $X$ on a probability space, $\sigma (X)$ is the smallest sigma-algebra making $X$ a measurable function into the Borel sigma-algebra on $\mathbb{R}$. The collection of all inverse images of Borel sets is already a sigma-algebra, since inverse images preserve complements and countable unions; this proves the displayed formula without completing any measure.

Let a random vector $V$ have a finite actual range of $q$ values. Its nonempty fibres partition the sample space, and every event in $\sigma (V)$ is a union of those fibres. Thus each fibre is an atom of a sigma-algebra and $\sigma (V)$ has exactly $2^q$ elements. This assertion concerns the actual range; additional values attained only on null sets must still be counted when working with an uncompleted sigma-algebra literally.

To verify the claim, the fibres belong to $\sigma (V)$ as preimages of singleton Borel sets. Their unions form a finite sigma-algebra making $V$ measurable, so minimality of the generated sigma-algebra gives the reverse inclusion. Distinct subcollections have distinct unions because the fibres are nonempty and disjoint.

Consequently a finite history of finite-valued observations generates a finite natural filtration at each fixed time. It may be strictly smaller than the filtration of its driving continuous observations: a single uniform random variable already generates infinitely many distinct events $\{U\le t\}$ with different probabilities.

A nonempty collection of subsets of a fixed universe closed under complements and finite unions, hence also finite intersections. It generates a sigma-algebra by allowing countable set operations. A countable generating algebra provides one simultaneous family of test indicators for measure-theoretic arguments.

A countable algebra of sets generating the intended sigma-algebra. Standard Borel spaces admit such algebras by taking finite Boolean combinations from a countable topological base. Finite linear combinations of its indicators are dense in $L^2$ for every Borel probability measure, by the Monotone class theorem. Countability permits a common full-measure convergence set for all test indicators.

A subset of $X$ is measurable relative to a sigma-algebra $\mathcal{F}$ when it belongs to $\mathcal{F}$. A measure defined on $\mathcal{F}$ assigns sizes to these sets. Measurability depends on the chosen sigma-algebra; for example, a Lebesgue measurable set need not be a Borel set.

A sigma-algebra atom is set-theoretically minimal. An atom is instead indivisible only modulo sets of measure zero.

An atom of a sigma-algebra is a nonempty measurable set containing no proper nonempty measurable subset. A finite sigma-algebra partitions its underlying set into atoms, and each measurable set is a union of them.

A Borel set in a topological space belongs to the smallest sigma-algebra containing every open set.

For a topological space, ${\Sigma }_1^0$ is the class of open sets and ${\Pi }_{\alpha }^0$ consists of complements of ${\Sigma }_{\alpha }^0$ sets. For countable $\alpha >1$, ${\Sigma }_{\alpha }^0$ consists of countable unions of sets in ${\bigcup }_{\beta <\alpha }{\Pi }_{\beta }^0$. These countable levels contain every Borel set. Continuous inverse images preserve each level, by induction using openness, complementation and countable unions.

The Borel sigma-algebra of a topological space $X$ is the sigma-algebra generated by its open sets. Its members are exactly the Borel sets.

Every subset of a discrete space is open, so its Borel sigma-algebra is the full power set.

A pi-system is a nonempty family of subsets closed under finite intersections.

A Dynkin system contains $\Omega$, is closed under complements, and is closed under countable disjoint unions. Equivalently, it contains differences $B\setminus A$ whenever $A\subseteq B$ are members.

Every Dynkin system containing a pi-system $\mathcal{P}$ also contains the sigma-algebra $\sigma (\mathcal{P})$. Equivalently, the Dynkin system generated by a pi-system equals the sigma-algebra it generates.