# monotone class theorem

Monotone Class theoremFernando Sanz Gamiz

###### Theorem.

Let ${\mathrm{F}}_{\mathrm{0}}$ an algebra of subsets of $\mathrm{\Omega}$. Let $\mathrm{M}$ be the smallest monotone class such that ${\mathrm{F}}_{\mathrm{0}}\mathrm{\subset}\mathrm{M}$ and $\sigma \mathit{}\mathrm{(}{\mathrm{F}}_{\mathrm{0}}\mathrm{)}$ be the sigma algebra generated by ${\mathrm{F}}_{\mathrm{0}}$. Then $\mathrm{M}\mathrm{=}\sigma \mathit{}\mathrm{(}{\mathrm{F}}_{\mathrm{0}}\mathrm{)}$.

###### Proof.

It is enough to prove that $\mathcal{M}$ is an algebra, because an algebra which is a monotone class is obviously a $\sigma $-algebra.

Let ${\mathcal{M}}_{A}=\{B\in \mathcal{M}|A\cap B,A\cap {B}^{\mathrm{\complement}}\text{and}{A}^{\mathrm{\complement}}\cap B\in \mathcal{M}\}$. Then is clear that ${\mathcal{M}}_{A}$ is a monotone class and, in fact, ${\mathcal{M}}_{A}=\mathcal{M}$, for if $A\in {\mathcal{F}}_{0}$, then ${\mathcal{F}}_{0}\subset {\mathcal{M}}_{A}$ since ${\mathcal{F}}_{0}$ is a field, hence $\mathcal{M}\subset {\mathcal{M}}_{A}$ by minimality of $\mathcal{M}$; consequently $\mathcal{M}={\mathcal{M}}_{A}$ by definition of ${\mathcal{M}}_{A}$. But this shows that for any $B\in \mathcal{M}$ we have $A\cap B,A\cap {B}^{\mathrm{\complement}}\text{and}{A}^{\mathrm{\complement}}\cap B\in \mathcal{M}$ for any $A\in {\mathcal{F}}_{0}$, so that ${\mathcal{F}}_{0}\subset {\mathcal{M}}_{B}$ and again by minimality $\mathcal{M}={\mathcal{M}}_{B}$. But what we have just proved is that $\mathcal{M}$ is an algebra, for if $A,B\in \mathcal{M}={\mathcal{M}}_{A}$ we have showed that $A\cap B,A\cap {B}^{\mathrm{\complement}}\text{and}{A}^{\mathrm{\complement}}\cap B\in \mathcal{M}$, and, of course, $\mathrm{\Omega}\in \mathcal{M}$. ∎

###### Remark 1.

One of the main applications of the Monotone Class Theorem is that of showing that certain property is satisfied by all sets in an $\sigma $-algebra, generally starting by the fact that the field generating the $\sigma $-algebra satisfies such property and that the sets that satisfies it constitutes a monotone class.

###### Example 1.

Consider an infinite^{} sequence^{} of independent random variables^{}
$\{{X}_{n},n\in \mathbb{N}\}$. The definition of independence is

$$P({X}_{1}\in {A}_{1},{X}_{2}\in {A}_{2},\mathrm{\dots},{X}_{n}\in {A}_{n})=P({X}_{1}\in {A}_{1})P({X}_{2}\in {A}_{2})\mathrm{\cdots}P({X}_{n}\in {A}_{n})$$ |

for any Borel sets ${A}_{1},{A}_{2},..,{A}_{n}$ and any finite $n$. Using the Monotone Class Theorem one can show, for example, that any event in $\sigma ({X}_{1},{X}_{2},\mathrm{\dots},{X}_{n})$ is independent of any event in $\sigma ({X}_{n+1},{X}_{n+2},\mathrm{\dots})$. For, by independence

$$P(({X}_{1},{X}_{2},\mathrm{\dots},{X}_{n})\in A,({X}_{n+1},{X}_{n+2},\mathrm{\dots})\in B)=P(({X}_{1},{X}_{2},\mathrm{\dots},{X}_{n})\in A)P(({X}_{n+1},{X}_{n+2},\mathrm{\dots})\in B)$$ |

when A and
B are measurable rectangles in ${\mathcal{B}}^{n}$ and ${\mathcal{B}}^{\mathrm{\infty}}$ respectively. Now it is clear that the sets A which
satisfies the above relation^{} form a monotone class. So

$$P(({X}_{1},{X}_{2},\mathrm{\dots},{X}_{n})\in A,({X}_{n+1},{X}_{n+2},\mathrm{\dots})\in B)=P(({X}_{1},{X}_{2},\mathrm{\dots},{X}_{n})\in A)P(({X}_{n+1},{X}_{n+2},\mathrm{\dots})\in B)$$ |

for every $A\in \sigma ({X}_{1},{X}_{2},\mathrm{\dots},{X}_{n})$ and any measurable rectangle $B\in {\mathcal{B}}^{\mathrm{\infty}}$. A second application of the theorem shows finally that the above relation holds for any $A\in \sigma ({X}_{1},{X}_{2},\mathrm{\dots},{X}_{n})$ and $B\in \sigma ({X}_{n+1},{X}_{n+2},\mathrm{\dots})$

Title | monotone class theorem |
---|---|

Canonical name | MonotoneClassTheorem |

Date of creation | 2013-03-22 17:07:34 |

Last modified on | 2013-03-22 17:07:34 |

Owner | fernsanz (8869) |

Last modified by | fernsanz (8869) |

Numerical id | 8 |

Author | fernsanz (8869) |

Entry type | Theorem |

Classification | msc 28A05 |

Related topic | MonotoneClass |

Related topic | SigmaAlgebra |

Related topic | Algebra |

Related topic | FunctionalMonotoneClassTheorem |