Essential range explained
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
be a measure space, and let
be a topological space. For any
({\calA},\sigma({\calT}))
-measurable
, we say the
essential range of
to mean the set
\operatorname{ess.im}(f)=\left\{y\inY\mid0<\mu(f-1(U))forallU\in{\calT}withy\inU\right\}.
[1] [2] [3] Equivalently,
\operatorname{ess.im}(f)=\operatorname{supp}(f*\mu)
, where
is the pushforward measure onto
of
under
and
\operatorname{supp}(f*\mu)
denotes the
support of
[4] Essential values
We sometimes use the phrase "essential value of
" to mean an element of the essential range of
[5] [6] Special cases of common interest
Y = C
Say
is
equipped with its usual topology. Then the essential range of
f is given by
\operatorname{ess.im}(f)=\left\{z\inC\midforall \varepsilon\inR>0:0<\mu\{x\inX:|f(x)-z|<\varepsilon\}\right\}.
[7] [8] [9] 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.
(Y,T) is discrete
Say
is
discrete, i.e.,
is the
power set of
i.e., the discrete topology on
Then the essential range of
f is the set of values
y in
Y with strictly positive
-measure:
\operatorname{ess.im}(f)=\{y\inY:0<\mu(fpre\{y\})\}=\{y\inY:0<(f*\mu)\{y\}\}.
[10] [11] [12] Properties
- The essential range of a measurable function, being the support of a measure, is always closed.
- The essential range ess.im(f) of a measurable function is always a subset of
\overline{\operatorname{im}(f)}
.
- The essential image cannot be used to distinguish functions that are almost everywhere equal: If
holds
-
almost everywhere, then
\operatorname{ess.im}(f)=\operatorname{ess.im}(g)
.
- These two facts characterise the essential image: It is the biggest set contained in the closures of
for all g that are a.e. equal to f:
\operatorname{ess.im}(f)=capf=ga.e.\overline{\operatorname{im}(g)}
.
- The essential range satisfies
\forallA\subseteqX:f(A)\cap\operatorname{ess.im}(f)=\emptyset\implies\mu(A)=0
.
- This fact characterises the essential image: It is the smallest closed subset of
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.
where f is considered as an element of the
C*-algebra
.
Examples
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
is open,
continuous and
the Lebesgue measure, then
\operatorname{ess.im}(f)=\overline{\operatorname{im}(f)}
holds. This holds more generally for all Borel measures that assign non-zero measure to every non-empty open set.
Extension
The notion of essential range can be extended to the case of
, where
is a separable metric space.If
and
are differentiable manifolds of the same dimension, if
VMO
and if
\operatorname{ess.im}(f)\neY
, then
.
[13] See also
References
Notes and References
- Book: Zimmer . Robert J. . Robert Zimmer . Essential Results of Functional Analysis . 1990 . University of Chicago Press . 0-226-98337-4 . 2.
- Book: Kuksin . Sergei . Sergei B. Kuksin . Shirikyan . Armen . 2012 . Mathematics of Two-Dimensional Turbulence . Cambridge University Press . 978-1-107-02282-9 . 292.
- Book: Kon . Mark A. . Probability Distributions in Quantum Statistical Mechanics . 1985 . Springer . 3-540-15690-9 . 74, 84.
- Book: Driver . Bruce . Analysis Tools with Examples . May 7, 2012 . 327 . Cf. Exercise 30.5.1.
- Book: Segal . Irving E. . Irving Segal . Kunze . Ray A. . Ray Kunze . Integrals and Operators . 1978 . Springer . 0-387-08323-5 . 106 . 2nd revised and enlarged.
- Book: Bogachev . Vladimir I. . Smolyanov . Oleg G. . Real and Functional Analysis . 2020 . Springer . 978-3-030-38219-3 . Moscow Lectures . 2522-0314 . 283.
- Book: Weaver . Nik . 2013 . Measure Theory and Functional Analysis . World Scientific . 978-981-4508-56-8 . 142.
- Book: Bhatia . Rajendra . Rajendra Bhatia . Notes on Functional Analysis . 2009 . Hindustan Book Agency . 978-81-85931-89-0 . 149.
- Book: Folland . Gerald B. . Gerald Folland . Real Analysis: Modern Techniques and Their Applications . 1999 . Wiley . 0-471-31716-0 . 187.
- Cf. Book: Tao . Terence . Terence Tao . Topics in Random Matrix Theory . 2012 . American Mathematical Society . 978-0-8218-7430-1 . 29.
- Cf. Book: Freedman . David . David A. Freedman . Markov Chains . 1971 . Holden-Day . 1.
- Cf. Book: Chung . Kai Lai . Chung Kai-lai . Markov Chains with Stationary Transition Probabilities . 1967 . Springer . 135.
- Brezis . Haïm . Nirenberg . Louis . Degree theory and BMO. Part I: Compact manifolds without boundaries . Selecta Mathematica . September 1995 . 1 . 2 . 197–263 . 10.1007/BF01671566.