---- The following standard two-step technique often works to prove that every set in a [[σ-algebra]] $\Sigma$ has a certain property: with $\mathcal{E}$ denoting the collection of sets satisfying the property, 1. Show that every element of a [[σ-algebra generated by a set collection|generating set]] $\mathscr{A}$ for $\Sigma$ satisfies the property, i.e., $\mathscr{A} \subset \mathcal{E}$. 2. Show that $\mathcal{E}$ is a $\sigma$-algebra, for then $\Sigma \subset \mathcal{E}$ since $\Sigma$ is the smallest $\sigma$-algebra containing $\mathscr{A}$. Often $(2)$ is very easy to check directly; sometimes it is not easy to check directly. The monotone class theorem is a useful tool for the latter case. See [[class methods in measure theory]] for more general discussion.[^1] > [!theorem] Theorem. ([[monotone class theorem for sets]]) > Suppose $\mathscr{A}$ is an [[algebra of sets|algebra]] on a set $W$. Then the [[σ-algebra]] [[σ-algebra generated by a set collection|generated by]] $\mathscr{A}$ equals the [[monotone class|monotone class generated by]] $\mathscr{A}$. ^theorem [^1]: See [[Dynkin's π-𝜆 theorem]] as well as the Canvas for [[class methods in measure theory.canvas|class methods in measure theory]]. > [!proof]- Proof. ([[monotone class theorem for sets]]) > Let $\sigma(\mathscr{A})$ denote the $\sigma$-algebra generated by $\mathscr{A}$ (i.e. the smallest $\sigma$-algebra containing $\mathscr{A}$). Let $\mathscr{M}(A)$ denote the monotone class generated by $\mathscr{A}$ (i.e. the smallest monotone class containing $\mathscr{A}$). Because every $\sigma$-algebra is a monotone class, $\mathscr{M}(\mathscr{A}) \subset \sigma(\mathscr{A})$. > > - [ ] the reverse inclusion (nothing clever so skipping for now, see handwritten notes) ---- #### ----- #### 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 > ```