Essential infimum and essential supremum explained

In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost everywhere, that is, except on a set of measure zero.

While the exact definition is not immediately straightforward, intuitively the essential supremum of a function is the smallest value that is greater than or equal to the function values everywhere while ignoring what the function does at a set of points of measure zero. For example, if one takes the function

f(x)

that is equal to zero everywhere except at

x=0

where

f(0)=1,

then the supremum of the function equals one. However, its essential supremum is zero if we apply the Lebesgue-Borel measure and are allowed to ignore what the function does at the single point where

is peculiar. The essential infimum is defined in a similar way.

Definition

As is often the case in measure-theoretic questions, the definition of essential supremum and infimum does not start by asking what a function

does at points

(that is, the image of

), but rather by asking for the set of points

where

equals a specific value

(that is, the preimage of

under

Let

f:X\to\Reals

be a real valued function defined on a set

The supremum of a function

is characterized by the following property:

f(x)\leq\supf\leqinfty

for all

x\inX

and if for some

a\in\Reals\cup\{+infty\}

we have

f(x)\leqa

for all

x\inX

then

\supf\leqa.

More concretely, a real number

is called an upper bound for

f(x)\leqa

for all

x\inX;

that is, if the set

f^(a, \infty) = \

is empty. Let

U_f = \\,

be the set of upper bounds of

and define the infimum of the empty set by

inf\varnothing=+infty.

Then the supremum of

\sup f = \inf U_f

if the set of upper bounds

U_f

is nonempty, and

\supf=+infty

otherwise.

Now assume in addition that

(X,\Sigma,\mu)

is a measure space and, for simplicity, assume that the function

is measurable. Similar to the supremum, the essential supremum of a function is characterised by the following property:

f(x)\leq\operatorname{ess}\supf\leqinfty

for

\mu

-almost all

x\inX

and if for some

a\in\Reals\cup\{+infty\}

we have

f(x)\leqa

for

\mu

-almost all

x\inX

then

\operatorname{ess}\supf\leqa.

More concretely, a number

is called an of

if the measurable set

f^-1(a,infty)

is a set of

\mu

-measure zero, That is, if

f(x)\leqa

for

\mu

-almost all

Let

U^_f = \

be the set of essential upper bounds. Then the is defined similarly as

\operatorname \sup f = \inf U^_f

U^{\operatorname{ess}

}_f \neq \varnothing, and

\operatorname{ess}\supf=+infty

otherwise.

Exactly in the same way one defines the as the supremum of the s, that is, $\operatorname \inf f = \sup \$ if the set of essential lower bounds is nonempty, and as

-infty

otherwise; again there is an alternative expression as

\operatorname{ess}inff=\sup\{a\in\Reals:f(x)\geqaforalmostallx\inX\}

(with this being

-infty

if the set is empty).

Examples

On the real line consider the Lebesgue measure and its corresponding -algebra

\Sigma.

Define a function

by the formula

f(x) = \begin5, & \text x=1 \\-4, & \text x = -1 \\2, & \text\end

The supremum of this function (largest value) is 5, and the infimum (smallest value) is −4. However, the function takes these values only on the sets

\{1\}

and

\{-1\},

respectively, which are of measure zero. Everywhere else, the function takes the value 2. Thus, the essential supremum and the essential infimum of this function are both 2.

As another example, consider the function $f(x) = \beginx^3, & \text x \in \Q \\\arctan x, & \text x \in \Reals \smallsetminus \Q \\\end$ where

denotes the rational numbers. This function is unbounded both from above and from below, so its supremum and infimum are

infty

and

-infty,

respectively. However, from the point of view of the Lebesgue measure, the set of rational numbers is of measure zero; thus, what really matters is what happens in the complement of this set, where the function is given as

\arctanx.

It follows that the essential supremum is

\pi/2

while the essential infimum is

-\pi/2.

On the other hand, consider the function

f(x)=x³

defined for all real

Its essential supremum is

+infty,

and its essential infimum is

-infty.

Lastly, consider the function $f(x) = \begin1/x, & \text x \neq 0 \\0, & \text x = 0. \\\end$ Then for any

a\in\Reals,

\mu(\{x\in\Reals:1/x>a\})\geq\tfrac{1}{|a|}

and so

	\operatorname{ess
U
	f

} = \varnothing and

\operatorname{ess}\supf=+infty.

Properties

\mu(X)>0

then

\inf f ~\leq~ \operatorname \inf f ~\leq~ \operatorname\sup f ~\leq~ \sup f.

and otherwise, if

has measure zero then ^[1]

+\infty ~=~ \operatorname\inf f ~\geq~ \operatorname\sup f ~=~ -\infty.

If the essential supremums of two functions

and

are both nonnegative, then

\operatorname{ess}\sup(fg)~\leq~(\operatorname{ess}\supf)(\operatorname{ess}\supg).

(S,\Sigma,\mu),

the space

l{L}^infty(S,\mu)

consisting of all of measurable functions that are bounded almost everywhere is a seminormed space whose seminorm

\|f\|_\infty = \inf \ = \begin\operatorname\sup |f| & \text 0 < \mu(S),\\0 & \text 0 = \mu(S),\end

is the essential supremum of a function's absolute value when

\mu(S) ≠ 0.

Notes and References

[Jean Dieudonné|Dieudonné J.]