Editing Independence (probability theory) (section)

===Independent σ-algebras===
The definitions above ({{EquationNote|Eq.1}} and {{EquationNote|Eq.2}}) are both generalized by the following definition of independence for [[sigma algebra|σ-algebras]]. Let <math>(\Omega, \Sigma, \mathrm{P})</math> be a probability space and let <math>\mathcal{A}</math> and <math>\mathcal{B}</math> be two sub-σ-algebras of <math>\Sigma</math>. <math>\mathcal{A}</math> and <math>\mathcal{B}</math> are said to be independent if, whenever <math>A \in \mathcal{A}</math> and <math>B \in \mathcal{B}</math>,

:<math>\mathrm{P}(A \cap B) = \mathrm{P}(A) \mathrm{P}(B).</math>

Likewise, a finite family of σ-algebras <math>(\tau_i)_{i\in I}</math>, where <math>I</math> is an [[index set]], is said to be independent if and only if

:<math>\forall \left(A_i\right)_{i\in I} \in \prod\nolimits_{i\in I}\tau_i \ : \ \mathrm{P}\left(\bigcap\nolimits_{i\in I}A_i\right) = \prod\nolimits_{i\in I}\mathrm{P}\left(A_i\right)</math>

and an infinite family of σ-algebras is said to be independent if all its finite subfamilies are independent.

The new definition relates to the previous ones very directly:
* Two events are independent (in the old sense) [[if and only if]] the σ-algebras that they generate are independent (in the new sense). The σ-algebra generated by an event <math>E \in \Sigma</math> is, by definition,
::<math>\sigma(\{E\}) = \{ \emptyset, E, \Omega \setminus E, \Omega \}.</math>
* Two random variables <math>X</math> and <math>Y</math> defined over <math>\Omega</math> are independent (in the old sense) if and only if the σ-algebras that they generate are independent (in the new sense). The σ-algebra generated by a random variable <math>X</math> taking values in some [[measurable space]] <math>S</math> consists, by definition, of all subsets of <math>\Omega</math> of the form <math>X^{-1}(U)</math>, where <math>U</math> is any measurable subset of <math>S</math>.

Using this definition, it is easy to show that if <math>X</math> and <math>Y</math> are random variables and <math>Y</math> is constant, then <math>X</math> and <math>Y</math> are independent, since the σ-algebra generated by a constant random variable is the trivial σ-algebra <math>\{ \varnothing, \Omega \}</math>. Probability zero events cannot affect independence so independence also holds if <math>Y</math> is only Pr-[[almost surely]] constant.