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A sequence of measurable functions  $  \{ f _ {i} \} $
such that

$$ 
\mu \{ {x } : {f _ {1} ( x) <
\alpha _ {1} \dots f _ {n} ( x) < \alpha _ {n} } \}
 = \ 
\prod _ {i=1} ^ { n } 
\mu \{ {x } : {f _ {i} ( x) < \alpha _ {i} } \}
$$

for any  $  n $
and any  $  \alpha _ {1} \dots \alpha _ {n} $. 
The simplest example of a system of independent functions is the [[Rademacher system|Rademacher system]].

(Kolmogorov's) criterion for the almost-everywhere convergence of a series of independent functions: For a series of independent functions  $  \sum _ {i=} 1  ^  \infty  f _ {i} $
to converge almost everywhere it is necessary and sufficient that for some  $  C > 0 $
the following three series converge:

$$ 
\sum _ { i } \mu
\{ {x } : {f _ {i} ( x) > C } \}
,\ \ 
\sum _ { i } \int\limits
f _ {i} ^ { C } ( x)  d x ,
$$

$$ 
\sum _ { i } \int\limits ( f _ {i} ^ { C } ( x) )  ^ {2} \ 
d x - \left ( \int\limits f _ {i} ^ { C } ( x)  d x \right )  ^ {2} ,
$$

where

$$ 
f _ {i} ^ { C } ( x)  = \ 
\left \{ 
\begin{array}{ll}
f _ {i} ( x) ,  & | f _ {i} ( x) | \leq  C ,  \\
0 ,  & | f _ {i} ( x) | > C .  \\
\end{array}

 \right .$$

====Comments====
Of course, to be able to introduce the concept of a system of independent functions one needs to have a [[measure space]]  $  ( X , \mu ) $
on which the functions are defined and measurable (with respect to  $  \mu $). 
Moreover,  $  \mu $
must be positive and finite, so  $  \mu $
can be taken a [[Probability measure|probability measure]] (then  $  ( X , \mu ) $
is a [[Probability space|probability space]]). An example is  $  ( X , \mu ) = ( [ 0 , 1 ],  \textrm{ Lebesgue  measure  } ) $.

In this abstract setting, instead of functions one takes random variables, thus obtaining a system of independent random variables.

The notion of a system of independent functions (random variables) should not be mixed up with that of an independent set of elements of a vector space  $  V $
over a field  $  K $: 
A set of elements  $  \{ x _ {1} \dots x _ {n} \} $
in  $  V $
such that for  $  c _ {i} \in K $, 
$  c _ {1} x _ {1} + \dots + c _ {n} x _ {n} = 0 $
implies  $  c _ {1} = \dots = c _ {n} = 0 $, 
see also [[Vector space|Vector space]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.-P. Kahane,   "Some random series of functions" , Cambridge Univ. Press  (1985)</TD></TR></table>