$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ # For sets A ==**monotone class**== is a collection of subsets of $\Omega$ which are closed under monotone limits $A_i\uparrow A$ and $A_i\downarrow A$. Take a look at [[pi lambda system|Dynkin's Theorem]], and prove the following results. >[!problem] Algebra $+$ MC $=\sigma$ >An algebra of sets of $\Omega$ is a $\sigma$-algebra iff it is a monotone class. (An algebra is closed under finite unions and complements). >[!solution]- > $\sigma$-algebra $\implies$ monotone class is trivial. To verify a monotone class algebra is a $\sigma$-algebra we need only verify countable unions. But this is trivial: given $(A_i)_{i\in \NN}$ create $B_i = \bigcup_{j\leq i} A_i$ and apply monotone class. >[!problem] >The intersection of an arbitrary family of monotone classes $\{\AAA_i\}_{i\in I}$ is a monotone class. >[!theorem] Monotone Class Theorem >For any algebra $\AAA$, the smallest monotone class containing $\AAA$ (that is, the union of all such) is precisely $\sigma(\AAA)$. >[!problem] Prove this > [!solution]- Fully analogous to Dynkin. > Let $\FFF$ be the union of all monotone classes containing $\AAA$. Show that all of the following are monotone classes containing $\AAA$$ > \FFF^{(1)} = \{A\in \mathcal{X}: \bar{A}\in \mathcal{X}\},\qquad \FFF^{(2)} = \{A\in \mathcal{X}: A\cup B\in \mathcal{X} \quad \forall B\in \AAA\} > $thus $\FFF = \FFF^{(1)} = \FFF^{(2)}$. Then, show$ > \FFF^{(3)} = \{A\in \mathcal{X}: A\cup B\in \mathcal{X} \quad \forall B\in \mathcal{X}\} > $is a monotone class containing $\AAA$, which concludes. # For Functions This is the more well-known one. >[!theorem] Monotone Class Theorem >Let $(\Omega, \FFF)$ be a measurable space and let $\AAA$ be a $\pi$-system generating $\FFF$. Let $V$ be a vector space of **bounded** functions $f:E\to \RR$ such that: >- $1\in V$ and $1_A\in V$ for all $A\in \AAA$. >- For any sequence of **nonnegative** $f_n\in V$, if $f_n\uparrow f$ is also bounded, then $f\in V$. > >Then, $V$ contains every bounded measurable function. See [[approximation by simple functions]].