Let
(
X
,
T
)
be a topological space, and let
Σ
be a σ-algebra on
X
that contains the topology
T
. (Thus, every open subset of
X
is a measurable set and
Σ
is at least as fine as the Borel σ-algebra on
X
.) Let
M
be a collection of (possibly signed or complex) measures defined on
Σ
. The collection
M
is called tight (or sometimes uniformly tight) if, for any
ε
>
0
, there is a compact subset
K
ε
of
X
such that, for all measures
μ
∈
M
,
|
μ
|
(
X
∖
K
ε
)
<
ε
.
where
|
μ
|
is the total variation measure of
μ
. Very often, the measures in question are probability measures, so the last part can be written as
μ
(
K
ε
)
>
1
−
ε
.
If a tight collection
M
consists of a single measure
μ
, then (depending upon the author)
μ
may either be said to be a tight measure or to be an inner regular measure.
If
Y
is an
X
-valued random variable whose probability distribution on
X
is a tight measure then
Y
is said to be a separable random variable or a Radon random variable.
If
X
is a metrisable compact space, then every collection of (possibly complex) measures on
X
is tight. This is not necessarily so for non-metrisable compact spaces. If we take
[
0
,
ω
1
]
with its order topology, then there exists a measure
μ
on it that is not inner regular. Therefore, the singleton
{
μ
}
is not tight.
If
X
is a compact Polish space, then every probability measure on
X
is tight. Furthermore, by Prokhorov's theorem, a collection of probability measures on
X
is tight if and only if it is precompact in the topology of weak convergence.
Consider the real line
R
with its usual Borel topology. Let
δ
x
denote the Dirac measure, a unit mass at the point
x
in
R
. The collection
M
1
:=
{
δ
n
|
n
∈
N
}
is not tight, since the compact subsets of
R
are precisely the closed and bounded subsets, and any such set, since it is bounded, has
δ
n
-measure zero for large enough
n
. On the other hand, the collection
M
2
:=
{
δ
1
/
n
|
n
∈
N
}
is tight: the compact interval
[
0
,
1
]
will work as
K
ε
for any
ε
>
0
. In general, a collection of Dirac delta measures on
R
n
is tight if, and only if, the collection of their supports is bounded.
Consider
n
-dimensional Euclidean space
R
n
with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures
Γ
=
{
γ
i
|
i
∈
I
}
,
where the measure
γ
i
has expected value (mean)
m
i
∈
R
n
and covariance matrix
C
i
∈
R
n
×
n
. Then the collection
Γ
is tight if, and only if, the collections
{
m
i
|
i
∈
I
}
⊆
R
n
and
{
C
i
|
i
∈
I
}
⊆
R
n
×
n
are both bounded.
Tightness and convergence
Tightness is often a necessary criterion for proving the weak convergence of a sequence of probability measures, especially when the measure space has infinite dimension. See
Finite-dimensional distribution
Prokhorov's theorem
Lévy–Prokhorov metric
weak convergence of measures
Tightness in classical Wiener space
Tightness in Skorokhod space
A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures
(
μ
δ
)
δ
>
0
on a Hausdorff topological space
X
is said to be exponentially tight if, for any
ε
>
0
, there is a compact subset
K
ε
of
X
such that
lim sup
δ
↓
0
δ
log
μ
δ
(
X
∖
K
ε
)
<
−
ε
.
