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 ![E[|X_n|^r]](/images/equations/ConvergenceinMean/Inline18.svg)
and ![E[|X|]](/images/equations/ConvergenceinMean/Inline19.svg)
all exist and if
![lim_(n->infty)E[|X_n-X|^r]=0](/images/equations/ConvergenceinMean/NumberedEquation2.svg) |
where ![E[Y]](/images/equations/ConvergenceinMean/Inline20.svg)
denotes the expectation value of 
. In this usage, convergence in the 
norm for the special case 
is called "convergence in mean."
