Convergence in mean denotes several forms of sequential convergence, depending on the mathematical context.
In functional analysis, convergence in mean is most often another name for strong convergence.
In particular, a sequence 
in a normed
space 
converges in mean to an element 
whenever
 |
as 
,
where 
denotes the norm on 
. Sometimes a sequence 
of functions in 
is said to converge in mean if 
converges in the 
norm to a function 
for a measure
space 
.
In probability and related theories, let 
and let the random
variables 
and 
be defined on a common probability space. The
sequence 
converges in the 
th mean (or, when 
, in the 
norm) to 
if the 
th absolute moments ![E[|X_n|^r]](/images/equations/ConvergenceinMean/Inline22.svg)
and ![E[|X|^r]](/images/equations/ConvergenceinMean/Inline23.svg)
are finite and
![lim_(n->infty)E[|X_n-X|^r]=0,](/images/equations/ConvergenceinMean/NumberedEquation2.svg) |
where ![E[Y]](/images/equations/ConvergenceinMean/Inline24.svg)
denotes the expectation value of 
. In this usage, convergence
in the 
norm for 
is called convergence in mean.
For every 
,
convergence in the 
th
mean implies probabilistic convergence.
Indeed, Markov's inequality gives ![P(|X_n-X|>epsilon)<=epsilon^(-r)E[|X_n-X|^r]](/images/equations/ConvergenceinMean/Inline30.svg)
for 
(Durrett 2019, p. 57).
It need not imply convergence of the entire sequence almost surely (Massachusetts Institute of Technology
2018, p. 12).
