The phrase "convergence in mean" is used in several branches of mathematics to refer to a number of different types of sequential convergence.
In functional analysis, "convergence in mean" is most often used as another name for strong
convergence. In particular, a sequence 
in a normed linear space 
converges in mean to an element 
whenever
 |
as 
,
where 
denotes the norm on 
. Sometimes, however, a sequence 
of functions in 
is said to converge in mean if 
converges in 
-norm to a function 
for some measure space 
.
The term is also used in probability and related theories to mean something somewhat different. In these contexts, a sequence 
of random
variables is said to converge in the 
th mean (or in the 
norm) to a random variable 
if the 
th absolute moments 
and 
all exist and if
 |
where 
denotes the expectation value. In this usage,
convergence in the 
norm for the special case 
is called "convergence in mean."
