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essential supremum

Essential supremum of a function

Let (Ω,ℱ,μ)normal-Ωℱμ(\Omega,\mathcal{F},\mu) be a measure space and let fff be a Borel measurable function from Ωnormal-Ω\Omega to the extended real numbers ℝ¯normal-¯ℝ\mathbb{\bar{R}}. The essential supremum of fff is the smallest number a∈ℝ¯anormal-¯ℝa\in\mathbb{\bar{R}} for which fff only exceeds aaa on a set of measure zero. This allows us to generalize the maximum of a function in a useful way.

More formally, we define ess⁢sup⁡fesssupremumf\mathrm{ess}\sup f as follows. Let a∈ℝaℝa\in\mathbb{R}, and define

Ma={x:f⁢(x)>a},subscriptMaconditional-setxfxaM_{{a}}=\{x:f(x)>a\},

the subset of XXX where f⁢(x)fxf(x) is greater than aaa. Then let

A0={a∈ℝ:μ⁢(Ma)=0},subscriptA0conditional-setaℝμsubscriptMa0A_{{0}}=\{a\in\mathbb{R}:\mu(M_{a})=0\},

the set of real numbers for which MasubscriptMaM_{a} has measure zero. The essential supremum of fff is

ess⁢sup⁡f:=inf⁡A0.assignesssupremumfinfimumsubscriptA0\mathrm{ess}\sup f:=\inf A_{0}.

The supremum is taken in the set of extended real numbers so, ess⁢sup⁡f=∞esssupremumf\mathrm{ess}\sup f=\infty if A0=∅subscriptA0A_{0}=\emptyset and ess⁢sup⁡f=-∞esssupremumf\mathrm{ess}\sup f=-\infty if A0=ℝsubscriptA0ℝA_{0}=\mathbb{R}.

Essential supremum of a collection of functions

Let (Ω,ℱ,μ)normal-Ωℱμ(\Omega,\mathcal{F},\mu) be a measure space, and 𝒮𝒮\mathcal{S} be a collection of measurable functions f:Ω→ℝ¯normal-:fnormal-→normal-Ωnormal-¯ℝf\colon\Omega\rightarrow\mathbb{\bar{R}}. The Borel σσ\sigma-algebra on ℝ¯normal-¯ℝ\mathbb{\bar{R}} is used.

If 𝒮𝒮\mathcal{S} is countable then we can define the pointwise supremum of the functions in 𝒮𝒮\mathcal{S}, which will itself be measurable. However, if 𝒮𝒮\mathcal{S} is uncountable then this is often not useful, and does not even have to be measurable. Instead, the essential supremum can be used.

The essential supremum of 𝒮𝒮\mathcal{S}, written as ess⁢sup⁡𝒮esssupremum𝒮{\mathrm{ess}\sup}\,\mathcal{S}, if it exists, is a measurable function f:Ω→ℝ¯normal-:fnormal-→normal-Ωnormal-¯ℝf\colon\Omega\rightarrow\mathbb{\bar{R}} satisfying the following.

  • f≥gfgf\geq g, μμ\mu-almost everywhere, for any g∈𝒮g𝒮g\in\mathcal{S}.

  • if g:Ω→ℝ¯normal-:gnormal-→normal-Ωnormal-¯ℝg\colon\Omega\rightarrow\mathbb{\bar{R}} is measurable and g≥hghg\geq h (μμ\mu-a.e.) for every h∈𝒮h𝒮h\in\mathcal{S}, then g≥fgfg\geq f (μμ\mu-a.e.).

Similarly, the essential infimum, ess⁢inf⁡𝒮essinfimum𝒮{\mathrm{ess}\inf}\mathcal{S} is defined by replacing the inequalities ‘≥\geq’ by ‘≤\leq’ in the above definition.

Note that if fff is the essential supremum and g:Ω→ℝ¯normal-:gnormal-→normal-Ωnormal-¯ℝg\colon\Omega\rightarrow\mathbb{\bar{R}} is equal to fff μμ\mu-almost everywhere, then ggg is also an essential supremum. Conversely, if f,gfgf,g are both essential supremums then, from the above definition, f≤gfgf\leq g and g≤fgfg\leq f, so f=gfgf=g (μμ\mu-a.e.). So, the essential supremum (and the essential infimum), if it exists, is only defined almost everywhere.

It can be shown that, for a σσ\sigma-finite measure μμ\mu, the essential supremum and essential infimum always exist. Furthermore, they are always equal to the supremum or infimum of some countable subset of 𝒮𝒮\mathcal{S}.

Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

Comments

Submitted by jujutsuka on Mon, 10/24/2005 - 17:07

why does this reduce to the usual notion of supremum when f is continuous?

Submitted by stevecheng on Tue, 10/25/2005 - 01:47

The set { x \in \R^n : f(x) > a }
is always open, and if it has measure zero at the same time,
then it is empty.

Submitted by bci1 on Sun, 02/01/2009 - 19:00

Should "essential supremum of a function" be capitalized in its first letter only, that is : "Essential supremum of a function"?
Same question about all the other subsections in this entry.
bci1

Submitted by gel on Sun, 02/01/2009 - 19:31

I'm not sure. I can't see anything in the content and style guide about this, although the guide itself does use capitals for all words in section headings and is not even consistent for subsections. I'll make it first word only for now.

Submitted by bci1 on Sun, 02/01/2009 - 19:44

Hello, gel:
That is exactly what I suggested: only the first letter of section subtitles in the text should be capitalized. Please use CWoo's entries as an example because he is the PMAdministrator handling such matters. The corrected subtitle I provided you with showed this.
Best wishes,
bci1

Submitted by gel on Sun, 02/01/2009 - 19:57

actually, I'm using sections and not subsections in my entries. Dunno which is preferable.

Submitted by bci1 on Sun, 02/01/2009 - 20:04

> actually, I'm using sections and not subsections in my
> entries. Dunno which is preferable.

The sections seem to be too large in the Noosphere implementation of Tex. Anyway, here I am sharing with you some of the instructions and examples provided to me by the PMAdmin, CC committee that you might also find useful about the `expected' PM style:

"some of the following entries are what we considered well-written entries:

two examples of a well-written definition:

http://planetmath.org/?op=getobj&from=objects&id=4844
http://planetmath.org/?op=getobj&from=objects&id=448

an example of a well-written theorem:

http://planetmath.org/?op=getobj&from=objects&id=3252

an example of a well-written proof:

http://planetmath.org/?op=getobj&from=objects&id=4463

an example of a well-written biographical/historical entry:

http://planetmath.org/?op=getobj&from=objects&id=4161

an example of a well-written topic entry:

http://planetmath.org/?op=getobj&from=objects&id=7592
http://planetmath.org/?op=getobj&from=objects&id=7007
http://planetmath.org/?op=getobj&from=objects&id=6878 "

http://planetmath.org/?op=getobj&from=objects&id=7592

topics on calculus (Topic)

This entry is an overview of many calculus related entries which can be found here, at PlanetMath.org. By calculus we mean real analysis at the high-school level or college level, and the entries in this page should be at either level. If an entry is written at a higher level, it will be indicated with a “GLâ€Â� tag.

Expositions
The following are books or notes on calculus:
An Introduction to Calculus.
: The real line
Basic properties of the real numbers:
Decimal expansion
Positive
Inverse number, Opposite number
(GL) Real number
Topic entry on real numbers
Functions of one real variable
Concept of function and real function
Computing limits
Limit of function
Limit of (sin x)/x as x tends to 0
Limit rules of functions
Improper limits
L'Hôpital's rule
Growth of exponential function
Example of computing limits using Taylor expansion
Continuity
Continuous and discontinuous functions
Continuity of natural power
Continuity of sine and cosine
Jump discontinuity example
Intermediate value theorem
Differentiation in one variable
Derivative
Sum rule
Power rule
Product rule
Quotient rule
Chain rule
Derivative of inverse function
Derivatives of sine and cosine
Bolzano's theorem
Least and greatest value of function
Higher order derivatives
Fractional Differentiation
Integration of functions of one real variable
Integral sign
Definition of Riemann integral
A lecture on integration by substitution
A lecture on integration by parts
A lecture on trigonometric integrals and trigonometric substitutions
A lecture on the partial fraction decomposition method
Definite integral
Definite integral
Riemann integral
Fundamental theorem of calculus
Improper integral
List of improper integrals
Approximate integration: Left Hand Rule
Right Hand Rule
Midpoint Rule
Trapezoidal Rule
Simpson's Rule

Integral equation
Fractional Integration
Integral Transforms
Fourier transform
Laplace transform
Fourier-Mellin integral
Mellin's transform
Fourier sine transform
Fourier cosine transform
Multivariable Calculus
Differentiation
Iterated limit
Jacobian matrix
Differentiation under the integral sign
Integration
Stokes' theorem
Differential Equations
Differential equation
(GL) Existence and uniqueness of solution of ordinary differential equations
Index of differential equations
Separation of variables
Method of integrating factors
(GL) Examples of solving a PDE: a) Heat equation, b) Wave equation
Infinite Series
Series of Numbers
Series
Topic entry on series of complex terms
Infinite Series: Tests for Convergence and Examples
(GL?) Non-existence of universal series convergence criterion
Cauchy general condition for convergence
Geometric series, Harmonic series
Sum of series depends on order
Manipulating convergent series
(GL?) Multiplication of series
Leibniz' estimate for alternating series
Function Sequences and Series
Limit of function sequence
The limit of a uniformly convergent sequence of continuous functions is continuous
Sum function of series
Termwise differentiation
Weierstrass' criterion of uniform convergence
(GL) Fourier series
Power Series and Taylor Series
Power series and Taylor series
Taylor's theorem
Newton's binomial series
Example of Taylor polynomials for
Example of Taylor polynomials for the exponential function
Examples on how to find Taylor series from other known series
Getting Taylor series from differential equation "

" This entry is an overview of many calculus related entries which can be found here, at PlanetMath.org. By calculus we \PMlinkescapetext{mean} real analysis at the high-school level or college level, and the entries in this page should be at either level. If an entry is written at a higher level, it will be indicated with a ``GL'' tag.

\section{Expositions}
The following are books or notes on calculus:
\begin{itemize}
\item \PMlinkexternal{An Introduction to Calculus}{http://planetmath.org/?op=getobj&from=lec&id=36}.
\end{itemize}

\section{$\Reals$ : The real line}
Basic properties of the real numbers:
\begin{itemize}
\item Decimal expansion
\item Positive
\item Inverse number, Opposite number
\item (GL) \PMlinkid{Real number}{RealNumber}
\item Topic entry on real numbers
\end{itemize}

\section{Functions of one real variable}
\begin{itemize}
\item Concept of function and real function
\end{itemize}

\section{Computing limits}
\begin{itemize}
\item Limit of function
\item \PMlinkname{Limit of (sin {\em x})/{\em x} as {\em x} tends to 0}{LimitOfDisplaystyleFracsinXxAsXApproaches0}
\item Limit rules of functions
\item Improper limits
\item \PMlinkname{L'H\^opital's rule}{LHpitalsRule}
\item Growth of exponential function
\item Example of computing limits using Taylor expansion
\end{itemize}

\section{Continuity}
\begin{itemize}
\item Continuous and discontinuous functions
\item Continuity of natural power
\item Continuity of sine and cosine
\item \PMlinkname{Jump discontinuity example}{ExampleOfJumpDiscontinuity}
\item Intermediate value theorem
\end{itemize}

\section{Differentiation in one variable}
\begin{itemize}
\item Derivative
\item Sum rule
\item Power rule
\item Product rule
\item Quotient rule
\item Chain rule
\item Derivative of inverse function
\item Derivatives of sine and cosine
\item Bolzano's theorem
\item Least and greatest value of function
\item Higher order derivatives
\item Fractional Differentiation
\end{itemize}

\section{Integration of functions of one real variable}
\begin{itemize}
\item Integral sign
\item \PMlinkname{Definition of Riemann integral}{RiemannIntegral}
\item A lecture on integration by substitution
\item A lecture on integration by parts
\item A lecture on trigonometric integrals and trigonometric substitutions
\item A lecture on the partial fraction decomposition method
\end{itemize}

\section{Definite integral}

\begin{itemize}
\item Definite integral
\item Riemann integral
\item Fundamental theorem of calculus
\item Improper integral
\item List of improper integrals
\item Approximate integration: \begin{tabular}{l}
Left Hand Rule \\
Right Hand Rule \\
Midpoint Rule \\
Trapezoidal Rule \\
Simpson's Rule \end{tabular}
\item Integral equation
\item Fractional Integration
\end{itemize}

\section{Integral Transforms}
\begin{itemize}
\item Fourier transform
\item Laplace transform
\item Fourier-Mellin integral
\item Mellin's transform
\item Fourier sine transform
\item Fourier cosine transform
\end{itemize}

\section{Multivariable Calculus}
\subsection{Differentiation}
\begin{itemize}
\item Iterated limit
\item Jacobian matrix
\item Differentiation under the integral sign
\end{itemize}

\subsection{Integration}
\begin{itemize}
\item Stokes' theorem
\end{itemize}

\section{Differential Equations}
\begin{itemize}
\item Differential equation
\item (GL) Existence and uniqueness of solution of ordinary differential equations
\item Index of differential equations
\item Separation of variables
\item Method of integrating factors
\item (GL) Examples of solving a PDE:\,
a) \PMlinkname{Heat equation}{ExampleOfSolvingTheHeatEquation},
b) \PMlinkname{Wave equation}{SolvingTheWaveEquationByDBernoulli}
\end{itemize}

\section{Infinite Series}

\subsection{Series of Numbers}
\begin{itemize}
\item Series
\item Topic entry on series of complex terms
\item \PMlinkexternal{Infinite Series: Tests for Convergence and Examples}{http://planetmath.org/?op=getobj&from=lec&id=37}
\item (GL?) Non-existence of universal series convergence criterion
\item \PMlinkname{Cauchy general condition for convergence}{CauchyCriterionForConvergence}
\item Geometric series, Harmonic series
\item Sum of series depends on order
\item Manipulating convergent series
\item (GL?) Multiplication of series
\item \PMlinkname{Leibniz' estimate for alternating series}{LeibnizEstimateForAlternatingSeries}
\end{itemize}

\subsection{Function Sequences and Series}
\begin{itemize}
\item Limit of function sequence
\item The limit of a uniformly convergent sequence of continuous functions is continuous
\item Sum function of series
\item Termwise differentiation
\item Weierstrass' criterion of uniform convergence
\item (GL) \PMlinkname{Fourier series}{FourierSineAndCosineSeries}
\end{itemize}

\subsection{Power Series and Taylor Series}
\begin{itemize}
\item Power series and Taylor series
\item Taylor's theorem
\item \PMlinkname{Newton's binomial series}{BinomialFormula}
\item \PMlinkid{Example of Taylor polynomials}{ExampleOfTaylorPolynomialsForSinX} for $\sin x$
\item Example of Taylor polynomials for the exponential function
\item Examples on how to find Taylor series from other known series
\item Getting Taylor series from differential equation
\end{itemize}
"

Submitted by gel on Sun, 02/01/2009 - 20:44

thanks. I'll have a look through those.
One comment on the first example though. It uses subsubsections with numbering, which is displayed as 0.0.1, 0.0.2 and looks a bit ugly, in my opinion. Numbering as 1, 2, etc would look better.
That's why I started using section rather than subsection in my entries, but that was before I realized that section*, subsection* suppresses the numbers anyway.

George

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Owner: gel
Added: 2002-02-17 - 19:16
Author(s): gel

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(v9) by gel 2013-03-22
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