Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 1, March 2022
Page(s) 35 - 41
DOI https://doi.org/10.1051/wujns/2022271035
Published online 16 March 2022

© Wuhan University 2022

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

A function z defined on a lattice and taking values in an Aabelian semigroup is called a valuation if(1)for all . A function z defined on some subset of L is called a valuation on if (1) holds whenever . For the set of convex bodies, , in with denoting union and intersection. Valuation on convex bodies is a classical concept. Probably the most famous result on valuations is Hadwiger’s classification theorem of continuous rigid motion invariant valuations[1]. For the more recent contributions on valuations on convex bodies readers can refer to Refs. [2-33].

Valuations on convex bodies can be considered as valuations on suitable function spaces. Recently, valuations on functions have been rapidly growing (see Refs. [34-55]). For a space of real-valued functions, the operations and are defined as pointwise maximum and minimum, respectively. A complete classification of valuations intertwining with the SL(n) on Sobolev space [46-49] and space [45,50-52,54] were established, respectively. Valuations on convex functions [34,36,39-42], quasi-concave functions [37,38], Lipschitz functions [43,44], and functions of Bounded variations [53] were introduced and classified.

Recently, Wang and Liu [55] showed that the Fourier transform is the only valuation which is a continuous, positive GL(n) covariant and logarithmic translation covariant complex-valued valuation on integral functions. This motivates the study of complex-valued valuations on functions.

Let L be a lattice of complex-valued functions. For , let and denote the real and imaginary parts of f, respectively. The pointwise maximum of f and g, and the pointwise minimum of f and g are defined by(2)and(3)If f, g are real-valued functions, then (2) and (3) coincide with the real cases. A function is called a valuation iffor all and if . It is called continuous ifIt is called translation invariant iffor every . It is called rotation invariant iffor every , where denotes the inverse of .

Let . If is a measure space, then the -space, is the collection of µ-measurable complex-valued functions that satisfiesA measure space is called non-atomic if for every with , there exists with and . Let denote the characteristic function of the measurable set E, i.e.

Theorem 1   Let be a non-atomic measure space and let be a continuous valuation. If there exist continuous functions with (k=1,2,3,4) such that for all and all with , then there exist constants such that for , andfor all . In addition, (k = 1,2,3,4) if .

Let denote the -space of Lebesgue measurable complex-valued functions on .

Theorem 2   A function is a continuous translation invariant valuation if and only if there exist continuous functions with the property that there exist constants such that for all (k = 1,2,3,4), and(4)for all .

Let be the unit sphere in and let denote the -space of spherical Lebesgue measurable complex-valued functions on .

Theorem 3   A function is a continuous rotation invariant valuation if and only if there exist continuous functions with the properties that and there exist constants such that for all (k = 1,2,3,4), and(5)for all .

1 Notation and Preliminary Results

We collect some properties of complex-valued functions. If f is a complex-valued function on , thenwhere and denote the real part and imaginary part of f, respectively. The absolute value of f which is also called modulus is defined byLet . For a measure space , define as the space of µ-measurable complex-valued functions that satisfiesLet , then . The functional is a semi-norm. If functions in that are equal almost everywhere with respect to are identified, then becomes a norm. Obviously, is a lattice of complex-valued functions. Let denote the subset of , where the functions take real values. For , if , then in . Moreover,

The following characterizations of real-valued valuations on functions which were established by Tsang [51] will play key role in our proof. Let and denote the space of Lebesgue measurable real-valued functions on and the space of spherical Lebesgue measurable real-valued functions on ,respectively.

Theorem 4  [51] A function is a continuous translation invariant valuation if and only if there exists a continuous function with the property that there exists a constant such that for all , andfor all .

Theorem 5  [51] A function is a continuous rotation invariant valuation if and only if there exists a continuous function with the properties that and there exist constants such that for all , andfor all .

2 Main Results

Lemma 1   Let be continuous functions with the properties that and there exist such that for all (k =1,...,4). If the function is defined by(6)then is a continuous valuation provided that if .

Proof   For , letBy (6) and (2), we obtainSimilarly, by (6) and (3), we haveNote that and that E, F, G, H are pairwise disjoints. Thus,Hence is a valuation.

It remains to show that is continuous. Let and let be a sequence in with in . Next, we will show that converges to by showing that every subsequence of has a subsequence, which converges to . Set and with such that and in . Let be a subsequence of , then converges to f in . Then there exists a subsequence of which converges to f in , where with such that and in . Since is continuous, we haveSince for all , we getIf , apply in to getAnd we take in the above equation if . By a modification of Lebesgue’s Dominated Convergence Theorem (see Ref.[51], Proposition 2.2), we have and(7)Similarly,(8)(9)(10)Thus,From (7)-(10), we conclude . Hence, is continuous.

Lemma 2    If the function is a valuation, thenfor all .

Proof   If or , then, by (2) and (3), we have(11)If or , then, by (2) and (3), we have(12)Note that , apply (11) and (12), then getfor all .

If we restrict f to , then it is obvious that is a valuation on . Also, we can construct another valuation on which is related to .

Lemma 3   Let be a valuation. If the functions is defined byfor all , then is a valuation on .

Proof   For , by (2) and (3), we have(13)By (13) and the valuation property of , it follows thatfor all . Thus, is a valuation on .

Lemma 4   Let be a valuation. If the functions are defined byfor all , then both are real-valued valuations on .

Proof   Since is a valuation, we havefor all . Thus,andfor all .

Therefore, both are real-valued valuations on .

In order to establish a representation theorem for continuous complex-valued valuations on , we will use the corresponding representation theorem for real case which was obtained by Tsang [51].

Theorem 6  [51] Let be a non-atomic measure space and let be a continuous translation invariant valuation. If there exists a continuous function with such that for all and all with , then there exist constants such that for all , andfor all . In addition, if .

Proof of Theorem 1  

Let be a continuous valuation. For , by Lemma 2, Lemma 3 and Lemma 4, we havewhere and .

Since . Moreover, Lemma 3 and Lemma 4 imply that are real-valued valuations on .

If we restrict to , then the continuity of implies that are continuous on .

If we consider with , then the continuity of implies that are continuous on . Thus, are continuous real-valued valuations on . It follows from Theorem 6 that there exist continuous functions with the properties that and there exist constants such that for all (k = 1,2,3,4), andfor all . In addition, (k = 1,2,3,4) if .

If µ is Lebesgue measure, then becomes the space of Lebesgue measurable complex-valued functions. We usually write as .

Lemma 5   Let be continuous functions with the property that there exists such that for all (k = 1,...,4). If the function is defined byThen is a continuous translation invariant valuation.

Proof    Let M enote the collection of Lebesgue measurable sets in . Take and µ Lebesgue measure in Lemma 1 to conclude that is a continuous valuation on .

For every and every , we havewhich means that is translation invariant.

Proof of Theorem 2  

It follows from Lemma 5 that (4) determines a continuous translation invariant valuation on . Conversely, let be a continuous translation invariant valuation. Taking and µ Lebesgue measure in the proof of Theorem 1, we obtainwhere are real-valued valuations on . Theorem 4 implies that there exist continuous functions with the property that there exist constants such that for all (k = 1,2,3,4), andfor all .

Let W denote the -algebra defined as

Also denote by the spherical Lebesgue measure. If µ is the spherical Lebesgue measure, then denotes the space of spherical Lebesgue measurable complex valued functions. We usually write as .

Lemma 6   Let be continuous functions with the properties that and there exists such that for all (k=1,...,4). If the function is defined bythen is a continuous rotation invariant valuation.

Proof   Take and in Lemma 1 to conclude that is a continuous valuation on .

Note that for every and every . Since the spherical Lebesgue measure is rotation invariant, we havefor all , which completes the proof.

Proof of Theorem 3  

It follows from Lemma 6 that (5) determines a continuous rotation invariant valuation on .

Conversely, let be a continuous rotation invariant valuation. Taking and in the proof of Theorem 1, we obtainwhere are real-valued valuations on .

Theorem 5 implies that there exist continuous functions with the properties that and there exist constants such that for all (k = 1,2,3,4), andfor all .

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