# independent sigma algebras

Let $(\mathrm{\Omega},\mathcal{B},P)$ be a probability space^{}. Let ${\mathcal{B}}_{1}$ and ${\mathcal{B}}_{2}$ be two sub sigma algebras of $\mathcal{B}$. Then ${\mathcal{B}}_{1}$ and ${\mathcal{B}}_{2}$ are said to be if for any pair of events ${B}_{1}\in {\mathcal{B}}_{1}$ and ${B}_{2}\in {\mathcal{B}}_{2}$:

$$P({B}_{1}\cap {B}_{2})=P({B}_{1})P({B}_{2}).$$ |

More generally, a finite set^{} of sub-$\sigma $-algebras ${\mathcal{B}}_{1},\mathrm{\dots},{\mathcal{B}}_{n}$ is *independent* if for any set of events ${B}_{i}\in {\mathcal{B}}_{i}$, $i=1,\mathrm{\dots},n$:

$$P({B}_{1}\cap \mathrm{\cdots}\cap {B}_{n})=P({B}_{1})\mathrm{\cdots}P({B}_{n}).$$ |

An arbitrary set $\mathcal{S}$ of sub-$\sigma $-algebras is *mutually independent* if any finite subset of $\mathcal{S}$ is independent.

The above definitions are generalizations^{} of the notions of independence (http://planetmath.org/Independent) for events and for random variables^{}:

- 1.
Events ${B}_{1},\mathrm{\dots},{B}_{n}$ (in $\mathrm{\Omega}$) are

mutually independentif the sigma algebras $\sigma ({B}_{i}):=\{\mathrm{\varnothing},{B}_{i},\mathrm{\Omega}-{B}_{i},\mathrm{\Omega}\}$ are mutually independent.- 2.
Random variables ${X}_{1},\mathrm{\dots},{X}_{n}$ defined on $\mathrm{\Omega}$ are

mutually independentif the sigma algebras ${\mathcal{B}}_{{X}_{i}}$ generated by (http://planetmath.org/MathcalFMeasurableFunction) the ${X}_{i}$’s are mutually independent.

In general, mutual independence among events ${B}_{i}$, random variables ${X}_{j}$, and sigma algebras ${\mathcal{B}}_{k}$ means the mutual independence among $\sigma ({B}_{i})$, ${\mathcal{B}}_{{X}_{j}}$, and ${\mathcal{B}}_{k}$.

Remark. Even when random variables ${X}_{1},\mathrm{\dots},{X}_{n}$ are defined on different probability spaces $({\mathrm{\Omega}}_{i},{\mathcal{B}}_{i},{P}_{i})$, we may form the product^{} (http://planetmath.org/InfiniteProductMeasure) of these spaces $(\mathrm{\Omega},\mathcal{B},P)$ so that ${X}_{i}$ (by abuse of notation) are now defined on $\mathrm{\Omega}$ and their independence can be discussed.

Title | independent sigma algebras |
---|---|

Canonical name | IndependentSigmaAlgebras |

Date of creation | 2013-03-22 16:22:58 |

Last modified on | 2013-03-22 16:22:58 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 9 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 60A05 |

Synonym | mutually independent $\sigma $-algebras |

Defines | mutually independent sigma algebras |