Convergence in measure of a sequence of finite-valued real or complex measurable functions 
to a finite-valued measurable
function 
of the same type on a measure space 
means that, for every 
,
![lim_(n->infty)mu([{x in X:|f_n(x)-f(x)|>epsilon]})=0.](/images/equations/ConvergenceinMeasure/NumberedEquation1.svg) |
When 
is a probability measure, this is probabilistic
convergence.
If 
,
almost everywhere convergence implies
convergence in measure. The finite-measure hypothesis cannot be omitted in general.
On 
with Lebesgue measure, let 
be the characteristic
function of ![[n,n+1]](/images/equations/ConvergenceinMeasure/Inline9.svg)
.
Then 
has pointwise convergence to 0, but the
measure of 
equals 1 for every 
and every 
. On an arbitrary measure space,
a sequence converging in measure has a subsequence
converging almost everywhere to the same function (Viaclovsky 2003, pp. 2-3).
