Infimum and supremum almost everywhere
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
that is equal to zero everywhere except at
where
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
be a real valued function defined on a set
The supremum of a function
is characterized by the following property:
for all
and if for some
we have
for all
then
More concretely, a real number
is called an upper bound for
if
for all
that is, if the set
![{\displaystyle f^{-1}(a,\infty )=\{x\in X:f(x)>a\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab37670f3496f1f242f52bc49033aa1e142b9ab0)
is empty. Let
![{\displaystyle U_{f}=\{a\in \mathbb {R} :f^{-1}(a,\infty )=\varnothing \}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2525607a59424b32f9344244d483081c6a9470e0)
be the set of upper bounds of
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
and define the infimum of the empty set by
![{\displaystyle \inf \varnothing =+\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9284d3b8b951d60b7b66bc8e03d6bde3668d81)
Then the supremum of
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is
![{\displaystyle \sup f=\inf U_{f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3e26712f9ba236128c9d9d80c28f1749d69227)
if the set of upper bounds
![{\displaystyle U_{f}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7be3c3c86d5d87a48bec069e8477dd5a4823fcc)
is nonempty, and
![{\displaystyle \sup f=+\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dff04596087e97971ede110712d47a10b7f4bb48)
otherwise.
Now assume in addition that
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:
for
-almost all
and if for some
we have
for
-almost all
then
More concretely, a number
is called an essential upper bound of
if the measurable set
is a set of
-measure zero,[a] That is, if
for
-almost all
in
Let
![{\displaystyle U_{f}^{\operatorname {ess} }=\{a\in \mathbb {R} :\mu (f^{-1}(a,\infty ))=0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60107a93a10c29823c392ef881c8b49c141badd1)
be the set of essential upper bounds. Then the
essential supremum is defined similarly as
![{\displaystyle \operatorname {ess} \sup f=\inf U_{f}^{\mathrm {ess} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7446eb869fc4c65642b454517e234b1f12769331)
if
![{\displaystyle U_{f}^{\operatorname {ess} }\neq \varnothing ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abd4a1bb9b03acb87171f6bc56d194673f14112a)
and
![{\displaystyle \operatorname {ess} \sup f=+\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/02bb73a838c028a33dc66a24938f428e3ff9125c)
otherwise.
Exactly in the same way one defines the essential infimum as the supremum of the essential lower bounds, that is,
![{\displaystyle \operatorname {ess} \inf f=\sup\{b\in \mathbb {R} :\mu (\{x:f(x)<b\})=0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc6c340c9740c91cbfffc587508dc95d1bbcf10)
if the set of essential lower bounds is nonempty, and as
![{\displaystyle -\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1)
otherwise; again there is an alternative expression as
![{\displaystyle \operatorname {ess} \inf f=\sup\{a\in \mathbb {R} :f(x)\geq a{\text{ for almost all }}x\in X\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14679cb5728a54457f0163c766343674dd81d3cf)
(with this being
![{\displaystyle -\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1)
if the set is empty).
Examples
On the real line consider the Lebesgue measure and its corresponding đ-algebra
Define a function
by the formula
![{\displaystyle f(x)={\begin{cases}5,&{\text{if }}x=1\\-4,&{\text{if }}x=-1\\2,&{\text{otherwise.}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1bbd001db6d3ed9d9cf9e86935923dbee63bc2)
The supremum of this function (largest value) is 5, and the infimum (smallest value) is â4. However, the function takes these values only on the sets
and
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
![{\displaystyle f(x)={\begin{cases}x^{3},&{\text{if }}x\in \mathbb {Q} \\\arctan x,&{\text{if }}x\in \mathbb {R} \smallsetminus \mathbb {Q} \\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab3311304510acc626f0945ce71a0bb1b9a04e3e)
where
![{\displaystyle \mathbb {Q} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a)
denotes the rational numbers. This function is unbounded both from above and from below, so its supremum and infimum are
![{\displaystyle \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21)
and
![{\displaystyle -\infty ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79d5ae78431259d0588140f827d0a1fcdfa4ec7f)
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
![{\displaystyle \arctan x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27b0552108b18b68df053891a2e0d4089a292379)
It follows that the essential supremum is
![{\displaystyle \pi /2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67)
while the essential infimum is
On the other hand, consider the function
defined for all real
Its essential supremum is
and its essential infimum is
Lastly, consider the function
![{\displaystyle f(x)={\begin{cases}1/x,&{\text{if }}x\neq 0\\0,&{\text{if }}x=0.\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d92d494e93578e84a163e401c2a4324457c2fa3)
Then for any
![{\displaystyle \mu (\{x\in \mathbb {R} :1/x>a\})\geq {\tfrac {1}{|a|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e755adc7e66b44f6744a26b4a45a1a28fd393591)
and so
![{\displaystyle U_{f}^{\operatorname {ess} }=\varnothing }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6854f06a288c8792b31d4fbba8b628b72a77e2)
and
Properties
If
then
![{\displaystyle \inf f~\leq ~\operatorname {ess} \inf f~\leq ~\operatorname {ess} \sup f~\leq ~\sup f.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f311d1c270b0b474780320dcbe42943901608b26)
and otherwise, if
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
has measure zero then
[1] ![{\displaystyle +\infty ~=~\operatorname {ess} \inf f~\geq ~\operatorname {ess} \sup f~=~-\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a092c23fec7f8eaf91c8516f9a0e1c75af587481)
If the essential supremums of two functions
and
are both nonnegative, then
Given a measure space
the space
consisting of all of measurable functions that are bounded almost everywhere is a seminormed space whose seminorm
![{\displaystyle \|f\|_{\infty }=\inf\{C\in \mathbb {R} _{\geq 0}:|f(x)|\leq C{\text{ for almost every }}x\}={\begin{cases}\operatorname {ess} \sup |f|&{\text{ if }}0<\mu (S),\\0&{\text{ if }}0=\mu (S),\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b105cce19b9940e5c566e37024bb8f91c5427fb)
is the essential supremum of a function's absolute value when
[nb 1] See also
Notes
- ^ For nonmeasurable functions the definition has to be modified by assuming that
is contained in a set of measure zero. Alternatively, one can assume that the measure is complete.
- ^ If
then
References
- ^ Dieudonné J.: Treatise On Analysis, Vol. II. Associated Press, New York 1976. p 172f.
This article incorporates material from Essential supremum on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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