In mathematics, particularly measure theory, the essential range of a function is intuitively the 'non-negligible' range of the function: It does not change between two functions that are equal almost everywhere. One way of thinking of the essential range of a function is the set on which the range of the function is most 'concentrated'. The essential range can be defined for measurable real or complex-valued functions on a measure space.
Let f be a Borel-measurable, complex-valued function defined on a measure space
(
X
,
A
,
μ
)
. Then the essential range of f is defined to be the set:
e
s
s
.
i
m
(
f
)
=
{
z
∈
C
∣
for all
ε
>
0
:
μ
(
{
x
:
|
f
(
x
)
−
z
|
<
ε
}
)
>
0
}
In other words: The essential range of a complex-valued function is the set of all complex numbers z such that the inverse image of each ε-neighbourhood of z under f has positive measure.
The essential range of a measurable function is always closed.
The essential range ess.im(f) of a measurable function is always a subset of
im
(
f
)
¯
.
The essential image cannot be used to distinguished functions that are almost everywhere equal: If
f
=
g
holds
μ
-almost everywhere, then
e
s
s
.
i
m
(
f
)
=
e
s
s
.
i
m
(
g
)
.
These two facts characterise the essential image: It is the biggest set contained in the closures of
im
(
g
)
for all g that are a.e. equal to f:
The essential range satisfies
∀
A
⊆
X
:
f
(
A
)
∩
e
s
s
.
i
m
(
f
)
=
∅
⟹
μ
(
A
)
=
0
.
This fact characterises the essential image: It is the smallest closed subset of
C
with this property.
The essential supremum of a real valued function equals the supremum of its essential image and the essential infimum equals the infimum of its essential range. Consequently, a function is essentially bounded if and only if its essential range is bounded.
The essential range of an essentially bounded function f is equal to the spectrum
σ
(
f
)
where f is considered as an element of the C*-algebra
L
∞
(
μ
)
.
If
μ
is the zero measure, then the essential image of all measurable functions is empty.
This also illustrates that even though the essential range of a function is a subset of the closure of the range of that function, equality of the two sets need not hold.
If
X
⊆
R
n
is open,
f
:
X
→
C
and
μ
the Lebesgue measure, then
e
s
s
.
i
m
(
f
)
=
im
(
f
)
¯
holds. This holds more generally for all Borel measures that assign non-zero measure to every non-empty open set.
