----- > [!proposition] Technique. ([[class methods in measure theory]]) > Suppose that $\mathcal{E}$ is the collection of all sets satisfying a particular property $\mathrm{P}$. How can one show that every set in a [[σ-algebra]] $\Sigma$ has the property $\mathrm{P}$? One should try the following options in order, proceeding to the next once once things get unwieldy or assumptions don't apply. > 1. Directly show $\Sigma \subset \mathcal{E}$. > 2. Show that $\mathscr{A} \subset \mathcal{E}$ for $\mathscr{A}$ a [[σ-algebra generated by a set collection|generating set]] of $\Sigma$, then show $\mathcal{E}$ is a $\sigma$-algebra.[^1] > 3. Show $\mathscr{A}$ is an [[algebra]] with $\mathscr{A} \subset \mathcal{E}$, [[monotone class theorem for sets|then show]] $\mathcal{E}$ is a [[monotone class]].[^2] > 4. Show $\mathscr{A} \subset \mathcal{E}$, then show $\mathscr{A}$ is a [[π-system]] and $\mathcal{E}$ is a [[Dynkin system|𝜆-system]].[^3] > ![[class methods in measure theory.canvas|class methods in measure theory]] ^the-property-playbook [^1]: For then $\Sigma \subset \mathcal{E}$ since $\Sigma$ is by definition the *smallest* $\sigma$-algebra containing $\mathscr{A}$. [^2]: For then $\Sigma=\sigma(\mathscr{A})=\operatorname{MonotoneClass}(\mathscr{A}) \subset \mathcal{E}$, using the [[monotone class theorem for sets|monotone class theorem]] (and the fact that $\text{MonotoneClass}(\mathscr{A})$ is by definition the *smallest* monotone class containing $\mathscr{A}$). [^3]: Indeed, if $\mathscr{A}=\mathcal{P}$ is a [[π-system]] and $\mathcal{E}=\mathcal{D}$ is a [[Dynkin system|𝜆-system]] such that $\mathcal{P} \subset \mathcal{D}$, then [[Dynkin's π-𝜆 theorem]] implies then $\Sigma=\sigma(\mathscr{A})=\sigma(\mathcal{P}) \subset \mathcal{D}=\mathcal{E}$. ----- #### ---- #### 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 > ```