Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability.
Let
f
,
f
n
(
n
∈
N
)
:
X
→
R
be measurable functions on a measure space (X,Σ,μ). The sequence (fn) is said to converge globally in measure to f if for every ε > 0,
lim
n
→
∞
μ
(
{
x
∈
X
:
|
f
(
x
)
−
f
n
(
x
)
|
≥
ε
}
)
=
0
,
and to converge locally in measure to f if for every ε > 0 and every
F
∈
Σ
with
μ
(
F
)
<
∞
,
lim
n
→
∞
μ
(
{
x
∈
F
:
|
f
(
x
)
−
f
n
(
x
)
|
≥
ε
}
)
=
0
.
Convergence in measure can refer to either global convergence in measure or local convergence in measure, depending on the author.
Throughout, f and fn (n
∈
N) are measurable functions X → R.
Global convergence in measure implies local convergence in measure. The converse, however, is false; i.e., local convergence in measure is strictly weaker than global convergence in measure, in general.
If, however,
μ
(
X
)
<
∞
or, more generally, if all the fn vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
If μ is σ-finite and (fn) converges (locally or globally) to f in measure, there is a subsequence converging to f almost everywhere. The assumption of σ-finiteness is not necessary in the case of global convergence in measure.
If μ is σ-finite, (fn) converges to f locally in measure if and only if every subsequence has in turn a subsequence that converges to f almost everywhere.
In particular, if (fn) converges to f almost everywhere, then (fn) converges to f locally in measure. The converse is false.
Fatou's lemma and the monotone convergence theorem hold if almost everywhere convergence is replaced by (local or global) convergence in measure.
If μ is σ-finite, Lebesgue's dominated convergence theorem also holds if almost everywhere convergence is replaced by (local or global) convergence in measure.
If X = [a,b] ⊆ R and μ is Lebesgue measure, there are sequences (gn) of step functions and (hn) of continuous functions converging globally in measure to f.
If f and fn (n ∈ N) are in Lp(μ) for some p > 0 and (fn) converges to f in the p-norm, then (fn) converges to f globally in measure. The converse is false.
If fn converges to f in measure and gn converges to g in measure then fn + gn converges to f + g in measure. Additionally, if the measure space is finite, fngn also converges to fg.
Let
X
=
R
, μ be Lebesgue measure, and f the constant function with value zero.
The sequence
f
n
=
χ
[
n
,
∞
)
converges to f locally in measure, but does not converge to f globally in measure.
The sequence
f
n
=
χ
[
j
2
k
,
j
+
1
2
k
]
where
k
=
⌊
log
2
n
⌋
and
j
=
n
−
2
k
(The first five terms of which are
χ
[
0
,
1
]
,
χ
[
0
,
1
2
]
,
χ
[
1
2
,
1
]
,
χ
[
0
,
1
4
]
,
χ
[
1
4
,
1
2
]
) converges to 0 globally in measure; but for no x does fn(x) converge to zero. Hence (fn) fails to converge to f almost everywhere.
The sequence
f
n
=
n
χ
[
0
,
1
n
]
converges to f almost everywhere and globally in measure, but not in the p-norm for any
p
≥
1
.
There is a topology, called the topology of (local) convergence in measure, on the collection of measurable functions from X such that local convergence in measure corresponds to convergence on that topology. This topology is defined by the family of pseudometrics
{
ρ
F
:
F
∈
Σ
,
μ
(
F
)
<
∞
}
,
where
ρ
F
(
f
,
g
)
=
∫
F
min
{
|
f
−
g
|
,
1
}
d
μ
.
In general, one may restrict oneself to some subfamily of sets F (instead of all possible subsets of finite measure). It suffices that for each
G
⊂
X
of finite measure and
ε
>
0
there exists F in the family such that
μ
(
G
∖
F
)
<
ε
.
When
μ
(
X
)
<
∞
, we may consider only one metric
ρ
X
, so the topology of convergence in finite measure is metrizable. If
μ
is an arbitrary measure finite or not, then
d
(
f
,
g
)
:=
inf
δ
>
0
μ
(
{
|
f
−
g
|
≥
δ
}
)
+
δ
still defines a metric that generates the global convergence in measure.
Because this topology is generated by a family of pseudometrics, it is uniformizable. Working with uniform structures instead of topologies allows us to formulate uniform properties such as Cauchyness.