Overview
Indpendence means that knowledge of whether one event has occurred does not convey any information about whether the other event has occurred.
Independent Events
An event A is independent of event B if the following holds
{% P(A) = P(A|B) = \frac{P(A \cap B)}{P(B)} %}
that is the probability of event A is the same as the probability of event A given the fact that B has occurred.
(B occurring confers no information about A)
this definition of independence implies that
{% P(A \cap B) = P(A)P(B) %}
Independent Random Variables
The defintion of independence of events can be used to define Independence of random variables In particular, the joint cumulative distribution function defines the probability over events in the joint probability space. As such, it can be used as the define independence of random variables as follows.
The variables {% X_1,X_2 ... X_n %} are independent if
{% F(X_1, ..., X_n) = F_{X_1}(x_1)F_{X_2}(x_2)...F_{X_n}(x_n) %}
Independence in Measure Theory
Sub-sigma algebras {% \mathcal{G}_1, ... ,\mathcal{G}_m %} of {% \mathcal{F} %} are independent if
{% \mathbb{P}(G_{1} \cap ... \cap G_{n}) = \Pi_i \mathbb{P}(G_{i}) %}
for
{% G_{i} \in \mathcal{G}_p %}
for some p.