Essential range

In mathematics, particularly measure theory, the essential range, or the set of essential values, 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 'concentrated'.

Formal definition

Let

(

X

,

A

,

μ

)

{\displaystyle (X,{\cal {A}},\mu )}

be a measure space, and let

(

Y

,

T

)

{\displaystyle (Y,{\cal {T}})}

be a topological space. For any

(

A

,

σ

(

T

)

)

{\displaystyle ({\cal {A}},\sigma ({\cal {T}}))}

-measurable function

f

:

X

→

Y

{\displaystyle f:X\to Y}

, we say the essential range of

f

{\displaystyle f}

to mean the set

e

s

s

.

i

m

⁡

(

f

)

=

{

y

∈

Y

∣

0

<

μ

(

f

−

1

(

U

)

)

for all

U

∈

T

with

y

∈

U

}

.

{\displaystyle \operatorname {ess.im} (f)=\left\{y\in Y\mid 0<\mu (f^{-1}(U)){\text{ for all }}U\in {\cal {T}}{\text{ with }}y\in U\right\}.}

Equivalently,

e

s

s

.

i

m

⁡

(

f

)

=

supp

⁡

(

f

∗

μ

)

{\displaystyle \operatorname {ess.im} (f)=\operatorname {supp} (f_{*}\mu )}

, where

f

∗

μ

{\displaystyle f_{*}\mu }

is the pushforward measure onto

σ

(

T

)

{\displaystyle \sigma ({\cal {T}})}

of

μ

{\displaystyle \mu }

under

f

{\displaystyle f}

and

supp

⁡

(

f

∗

μ

)

{\displaystyle \operatorname {supp} (f_{*}\mu )}

denotes the support of

f

∗

μ

.

{\displaystyle f_{*}\mu .}

Essential values

The phrase "essential value of

f

{\displaystyle f}

" is sometimes used to mean an element of the essential range of

f

.

{\displaystyle f.}

Special cases of common interest

Y = C

Say

(

Y

,

T

)

{\displaystyle (Y,{\cal {T}})}

is

C

{\displaystyle \mathbb {C} }

equipped with its usual topology. Then the essential range of f is given by

e

s

s

.