---- > [!definition] Definition. ([[σ-algebra]]) > A **$\sigma$-algebra** on a set $X$ is an [[algebra of sets|algebra]] on $X$ that is stable not only under finite unions, but [[countably infinite|countable]] unions as well. > Explicitly, a **$\sigma$-algebra** on $X$ is a nonempty collection $\Sigma \subset 2^{X}$ of subsets, called the **($\Sigma$-)measurable (sub)sets**, satisfying > > 1. $\emptyset \in \Sigma$; > 2. *(complements)* If $E \in \Sigma$, then $X-E \in \Sigma$; > 3. *(countable unions)* If $E_{1},E_{2}, \dots$ is a sequence of elements in $\Sigma$, then $\bigcup_{k=1}^{\infty}E_{k} \in \Sigma$. > > A **measurable space** is a pair $(X, \Sigma)$ where $X$ is a set and $\Sigma$ is a $\sigma$-algebra on $X$. > > ![[class methods in measure theory.canvas|class methods in measure theory]] > [!equivalence] > $\Sigma$ is a $\sigma$-algebra if and only if it is both a [[Dynkin system|𝜆-system]] and a [[π-system]]. ^equivalence > [!basicproperties] > > Suppose $\Sigma$ is a $\sigma$-algebra on a set $X$. Then > 1. $X \in \Sigma$; > 2. If $D,E \in S$ then $D - E \in \Sigma$ > 3. *(countable intersections)* If $E_{1}, E_{2},\dots$ is a sequence of elements in $\Sigma$, then $\bigcap_{k=1}^{\infty}E_{k} \in \Sigma$. > > > **1.** Applying defining property (2), then (1), this is clear. > > **2.** Noting that $D-E=D \cap (X-E)$, this is clear. > > **3.** Per [[De Morgan's Laws]], $X-\bigcap_{k=1}^{\infty}E_{k}= \bigcup_{k=1}^{\infty}(X-E_{k})$. By applying defining property (2) and then (3) to the RHS we see this belongs to $\Sigma$. Thus so does its complement $\bigcap_{k=1}^{\infty}E_{k}$. > [!basicexample] > > > - The [[discrete topology|trivial topology]] $\{ \emptyset, X \}$ is clearly a $\sigma$-algebra for any set $X$, as is the [[discrete topology]] $2^{X}$. (We can accordingly call these the **trivial $\sigma$-algebra** and **discrete $\sigma$-algebra**. It is not generally useful to compare the notions of $\sigma$-algebra and [[topological space|topology]], though.) > - The set of subsets $E$ of a set $X$ such that $E$ is [[countably infinite|countable]] or $X-E$ is [[countably infinite|countable]] forms a $\sigma$-algebra on $X$, as follows from the fact that the [[countable union of countable sets is countable]]. > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```