© Wuhan University 2022

0 Introduction

A function z defined on a lattice $(L,\vee ,\wedge )$ and taking values in an Aabelian semigroup is called a valuation if$z(f\vee g)+z(f\wedge g)=z(f)+z(g)$(1)for all $f,g\in L$. A function z defined on some subset $L_0$ of L is called a valuation on $L_0$ if (1) holds whenever $f,g,f\vee g,f\wedge g\in L_0$. For $L_0$ the set of convex bodies, $K^n$, in $R^n$ with $\vee$ denoting union and $\wedge$ 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 $\vee$ and $\wedge$ are defined as pointwise maximum and minimum, respectively. A complete classification of valuations intertwining with the SL(n) on Sobolev space ^[46-49] and $L^p$ 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 $f\in L$, let $\Re f$ and $\Im f$ denote the real and imaginary parts of f, respectively. The pointwise maximum of f and g, $f\vee g$ and the pointwise minimum of f and g$f\wedge g$ are defined by$f\vee g=\Re f\vee \Re g+\text{i}(\Im f\vee \Im g)$(2)and$f\wedge g=\Re f\wedge \Re g+\text{i}(\Im f\wedge \Im g)$(3)If f, g are real-valued functions, then (2) and (3) coincide with the real cases. A function $\Phi :L\to C$ is called a valuation if$\Phi (f\vee g)+\Phi (f\wedge g)=\Phi (f)+\Phi (g)$for all $f,g\in L$ and $\Phi (0)=0$ if $0\in L$. It is called continuous if$\Phi (f_i)\to \Phi (f),\text{ as }{f}_i\to f\text{ in }L\text{.}$It is called translation invariant if$\Phi (f(\cdot -t))=\Phi (f)$for every $t\in R^n$. It is called rotation invariant if$\Phi (f\circ {\theta }^{-1})=\Phi (f)$for every $\theta \in {\rm O}(n)$, where ${\theta }^{-1}$ denotes the inverse of $\theta$.

Let $p\ge 1$. If $(X,\aleph ,\mu )$ is a measure space, then the $L^p$-space, $L^p(C,\mu )$ is the collection of µ-measurable complex-valued functions $f:X\to C$ that satisfies${\displaystyle {\int }_X{\left|f\right|}^p\text{d}\mu }\wedge \infty$A measure space $(X,\aleph ,\mu )$ is called non-atomic if for every $E\in \aleph$ with $\mu (E)\vee 0$, there exists $F\in \aleph$ with $F\subseteq E$ and $0<\mu (F)<\mu (E)$. Let ${\chi }_E$ denote the characteristic function of the measurable set E, i.e.${\chi }_E=\{\begin{array}{l}1,x\in E\\ 0,x\notin E\end{array}$

Theorem 1  　Let $(X,\aleph ,\mu )$ be a non-atomic measure space and let $\Phi :L^p(C,\mu )\to C$ be a continuous valuation. If there exist continuous functions $h_k:R\to R$ with $h_k(0)=0$(k=1,2,3,4) such that $\Phi (c{\chi }_E)=$$(h_1(\Re c)+h_3(\Im c))\mu (E)+\text{i}(h_2(\Re c)+h_4(\Im c))\mu (E)$ for all $c\in C$ and all $E\in \aleph$ with $\mu (E)\wedge \infty$, then there exist constants ${\gamma }_k,{\delta }_k\ge 0$ such that $\left|h_k(a)\right|\le {\gamma }_k{\left|a\right|}^p$$+{\delta }_k$ for $a\in R$, and$\begin{array}{c}\Phi (f)={\displaystyle {\int }_X(h_1\circ \Re f+h_3\circ \Im f)}\text{d}\mu \\ +\text{i}{\displaystyle {\int }_X(h_2\circ \Re f+h_4\circ \Im f)}\text{d}\mu \end{array}$for all $f\in L^p(C,\mu )$. In addition, ${\delta }_k=0$(k = 1,2,3,4) if $\mu (X)=\infty$.

Let $L^p(C,R^n)$ denote the $L^p$-space of Lebesgue measurable complex-valued functions on $R^n$.

Theorem 2  　A function $\Phi :L^p(C,R^n)\to C$ is a continuous translation invariant valuation if and only if there exist continuous functions $h_k:R\to R$ with the property that there exist constants ${\gamma }_k\ge 0$ such that $|h_k(a)|\le {\gamma }_k|a{|}^p$ for all $a\in R$(k = 1,2,3,4), and$\begin{array}{c}\Phi (f)={\displaystyle {\int }_{R^n}(h_1\circ \Re f+h_3\circ \Im f)(x)}\text{d}x\\ +\text{i}{\displaystyle {\int }_{R^n}(h_2\circ \Re f+h_4\circ \Im f)(x)}\mathrm{d}x\end{array}$(4)for all $f\in L^p(C,R^n)$.

Let $S^{n-1}$ be the unit sphere in $R^n$ and let $L^p(C,S^{n-1})$ denote the $L^p$-space of spherical Lebesgue measurable complex-valued functions on $S^{n-1}$.

Theorem 3  　A function $\Phi :L^p(C,S^{n-1})\to C$ is a continuous rotation invariant valuation if and only if there exist continuous functions $h_k:R\to R$ with the properties that $h_k(0)=0$ and there exist constants ${\gamma }_k,{\delta }_k\ge 0$ such that $\left|h_k(a)\right|\le {\gamma }_k{\left|a\right|}^p+{\delta }_k$ for all $a\in R$(k = 1,2,3,4), and$\begin{array}{c}\Phi (f)={\displaystyle {\int }_{S^{n-1}}(h_1\circ \Re f+h_3\circ \Im f)}(u)\text{d}u\\ +\text{i}{\displaystyle {\int }_{S^{n-1}}(h_2\circ \Re f+h_4\circ \Im f)(u)}\text{d}u\end{array}$(5)for all $f\in L^p(C,S^{n-1})$.

1 Notation and Preliminary Results

We collect some properties of complex-valued functions. If f is a complex-valued function on $R^n$, then$f\left(x\right)=\Re f+\text{i}\Im f$where $\Re f$ and $\Im f$ denote the real part and imaginary part of f, respectively. The absolute value of f which is also called modulus is defined by$\left|f\right|=\sqrt{{(\Re f)}^2+{(\Im f)}^2}$Let $p\ge 1$. For a measure space $(X,\aleph ,\mu )$, define $L^p(C,\mu )$ as the space of µ-measurable complex-valued functions $f:X\to C$ that satisfies${\Vert f\Vert }_p={\left({\displaystyle {\int }_X{\left|f\right|}^p\text{d}\mu }\right)}^{\frac{1}{p}}\wedge \infty$Let $f,g\in L^p(C,\mu )$, then $f\vee g,f\wedge g\in L^p(C,\mu )$. The functional $\left|\right|\cdot \left|\right|:L^p(C,\mu )\to R$ is a semi-norm. If functions in $L^p(C,\mu )$ that are equal almost everywhere with respect to $\mu (\text{a}.\text{e}.[\mu ])$ are identified, then $\Vert \cdot \Vert :L^p(C,\mu )\to R$ becomes a norm. Obviously, $L^p(C,\mu )$ is a lattice of complex-valued functions. Let $L^p(R,\mu )$ denote the subset of $L^p(C,\mu )$, where the functions take real values. For $f_i,f\in L^p(C,\mu )$, if ${\Vert f_i-f\Vert }_p\to 0$, then $f_i\to f$ in $L^p(C,\mu )$. Moreover,$\begin{array}{l}{f}_i\to f\in L^p(C,\mu )\iff \Re f_i\to \Re f,\Im f_i\to \Im f\in \\ L^p(R,\mu )\end{array}$

The following characterizations of real-valued valuations on functions which were established by Tsang ^[51] will play key role in our proof. Let $L^p(R,R^n)$ and $L^p(R,S^{n-1})$ denote the $L^p$ space of Lebesgue measurable real-valued functions on $R^n$ and the $L^p$ space of spherical Lebesgue measurable real-valued functions on $S^{n-1}$，respectively.

Theorem 4  ^[51]　A function $\Phi :L^p(R,R^n)\to R$ is a continuous translation invariant valuation if and only if there exists a continuous function $h:R\to R$ with the property that there exists a constant $\gamma \ge 0$ such that $\left|h(a)\right|\le \gamma {\left|a\right|}^p$ for all $a\in R$, and$\Phi \left(f\right)={\displaystyle {\int }_{R^n}(h\circ f)(x)\text{d}x}$for all $f\in L^p(R,R^n)$.

Theorem 5  ^[51]　A function $\Phi :L^p(R,S^{n-1})\to R$ is a continuous rotation invariant valuation if and only if there exists a continuous function $h:R\to R$ with the properties that $h(0)=0$ and there exist constants $\gamma ,\delta \ge 0$ such that $\left|h(a)\right|\le \gamma {\left|a\right|}^p+\delta$ for all $a\in R$, and$\Phi (f)={\displaystyle {\int }_{S^{n-1}}\left(h\circ f\right)(u)\text{d}u}$for all $f\in L^p(R,S^{n-1})$.

2 Main Results

Lemma 1  　Let $h_k:R\to R$ be continuous functions with the properties that $h_k(0)=0$ and there exist ${\gamma }_k,{\delta }_k\ge 0$ such that $\left|h_k(a)\right|\le {\gamma }_k{\left|a\right|}^p+{\delta }_k$ for all $a\in R$(k =1,...,4). If the function $\Phi :L^p(C,\mu )\to C$ is defined by$\begin{array}{r}\Phi (f)={\displaystyle {\int }_X(h_1\circ \Re f+h_3\circ \Im f)}\text{d}\mu \\ +\text{i}{\displaystyle {\int }_X(h_2\circ \Re f+h_4\circ \Im f)}\text{d}\mu \end{array}$(6)then $\Phi$ is a continuous valuation provided that ${\delta }_k=0$ if $\mu (X)=\infty$.

Proof  　For $f,g\in L^p(C,\mu )$, let$\begin{array}{l}E=\left\{x\in X:\Re f\le \Re g,\Im f\le \Im g\right\},\\ F=\left\{x\in X:\Re f\le \Re g,\Im f\vee \Im g\right\},\\ G=\left\{x\in X:\Re f\vee \Re g,\Im f\le \Im g\right\},\\ H=\left\{x\in X:\Re f\vee \Re g,\Im f\vee \Im g\right\}.\end{array}$By (6) and (2), we obtain$\begin{array}{l}\Phi (f\vee g)\\ ={\displaystyle {\int }_X(h_1\circ \Re (f\vee g)+h_3\circ \Im (f\vee g))}\text{d}\mu \\ +\text{i}{\displaystyle {\int }_X(h_2\circ \Re (f\vee g)+h_4\circ \Im (f\vee g))}\text{d}\mu \\ ={\displaystyle {\int }_E(h_1\circ \Re g+h_3\circ \Im g)}\text{d}\mu +\text{i}{\displaystyle {\int }_E(h_2\circ \Re g\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{h}_4\circ \Im g)}\text{d}\mu \\ +{\displaystyle {\int }_F(h_1\circ \Re g+h_3\circ \Im f)}\text{d}\mu +\text{i}{\displaystyle {\int }_F(h_2\circ \Re g+h_4\circ \Im f)}\text{d}\mu \\ +{\displaystyle {\int }_G(h_1\circ \Re f+h_3\circ \Im g)}\text{d}\mu +\text{i}{\displaystyle {\int }_G(h_2\circ \Re f+h_4\circ \Im g)}\text{d}\mu \\ +{\displaystyle {\int }_H(h_1\circ \Re f+h_3\circ \Im f)}\text{d}\mu +\text{i}{\displaystyle {\int }_H(h_2\circ \Re f+h_4\circ \Im f)}\text{d}\mu \end{array}$Similarly, by (6) and (3), we have$\begin{array}{l}\Phi (f\wedge g)\\ ={\displaystyle {\int }_E(h_1\circ \Re f+h_3\circ \Im f)}\text{d}\mu +\text{i}{\displaystyle {\int }_E(h_2\circ \Re f+h_4\circ \Im f)}\text{d}\mu \\ +{\displaystyle {\int }_F(h_1\circ \Re f+h_3\circ \Im g)}\text{d}\mu +\text{i}{\displaystyle {\int }_F(h_2\circ \Re f+h_4\circ \Im g)}\text{d}\mu \\ +{\displaystyle {\int }_G(h_1\circ \Re g+h_3\circ \Im f)}\text{d}\mu +\text{i}{\displaystyle {\int }_G(h_2\circ \Re g+h_4\circ \Im f)}\text{d}\mu \\ +{\displaystyle {\int }_H(h_1\circ \Re g+h_3\circ \Im g)}\text{d}\mu +\text{i}{\displaystyle {\int }_H(h_2\circ \Re g+h_4\circ \Im g)}\text{d}\mu \end{array}$Note that $E\cup F\cup G\cup H=X$ and that E, F, G, H are pairwise disjoints. Thus,$\Phi (f\vee g)+\Phi (f\wedge g)=\Phi (f)+\Phi (g)$Hence $\Phi$ is a valuation.

It remains to show that $\Phi$ is continuous. Let $f\in L^p(C,\mu )$ and let $\left\{f_k\right\}$ be a sequence in $L^p(C,\mu )$ with $f_k\to f$ in $L^p(C,\mu )$. Next, we will show that $\Phi (f_k)$ converges to $\Phi (f)$ by showing that every subsequence $\Phi (f_{k_l})$ of $\Phi (f_{k_l})$ has a subsequence, $\Phi (f_{k_{l_m}})$ which converges to $\Phi (f)$. Set $f=\alpha +\text{i}\beta$ and $f={\alpha }_k+\text{i}{\beta }_k$ with $\alpha ,\beta ,{\alpha }_k,$${\beta }_k\in L^p(R,\mu )$ such that ${\alpha }_k\to \alpha$ and ${\beta }_k\to \beta$ in $L^p(R,\mu )$. Let $\left\{f_{k_l}\right\}$ be a subsequence of $\left\{f_k\right\}$, then $\left\{f_{k_l}\right\}$ converges to f in $L^p(C,\mu )$. Then there exists a subsequence $\left\{f_{k_{l_m}}\right\}$ of $\left\{f_{k_l}\right\}$ which converges to f in $L^p(C,\mu )$, where $f_{k_{l{}_m}}={\alpha }_{k_{l_m}}+\text{i}{\beta }_{k_{l_m}}$ with ${\alpha }_{k_{l_m}},{\beta }_{k_{l_m}}\in L^p(R,\mu )$ such that ${\alpha }_{k_{l_m}}\to \alpha$ and ${\beta }_{k_{l_m}}\to \beta$ in $L^p(R,\mu )$. Since $h_1$ is continuous, we have$(h_1\circ {\alpha }_{k_{l_m}})(x)\to (h_1\circ \alpha )(x),\text{\hspace{0.17em}a.e}.[\mu ]$Since $\left|h_1(a)\right|\le {\gamma }_1{\left|a\right|}^p+{\delta }_1$ for all $a\in R$, we get$\left|(h_1\circ {\alpha }_{k_{l_m}})(x)\right|\le {\gamma }_1{\left|{\alpha }_{k_{l_m}}\right|}^p+{\delta }_1,\text{\hspace{0.17em}a.e}.[\mu ].$If $\mu (x)\wedge \infty$, apply ${\alpha }_{k_{l_m}}\to \alpha$ in $L^p(R,\mu )$ to get$\underset{m\to \infty }{\lim }{\displaystyle {\int }_X{\gamma }_1{\left|{\alpha }_{k_{l_m}}\right|}^p+{\delta }_1\text{d}\mu =}{\displaystyle {\int }_X{\gamma }_1{\left|\alpha \right|}^p\text{d}\mu +{\delta }_1}\mu (X)$And we take ${\delta }_1=0$ in the above equation if $\mu (x)=\infty$. By a modification of Lebesgue’s Dominated Convergence Theorem (see Ref.[51], Proposition 2.2), we have $h_1\circ (\alpha )\in L^1(R,\mu )$ and$\underset{m\to \infty }{\lim }{\displaystyle {\int }_X(h_1\circ {\alpha }_{k_{l_m}})\text{d}\mu =}{\displaystyle {\int }_X(h_1\circ \alpha )\text{d}\mu }$(7)Similarly,$\underset{m\to \infty }{\lim }{\displaystyle {\int }_X(h_2\circ {\alpha }_{k_{l_m}})\text{d}\mu =}{\displaystyle {\int }_X(h_2\circ \alpha )\text{d}\mu }$(8)$\underset{m\to \infty }{\lim }{\displaystyle {\int }_X(h_3\circ {\beta }_{k_{l_m}})\text{d}\mu =}{\displaystyle {\int }_X(h_3\circ \beta )\text{d}\mu }$(9)$\underset{m\to \infty }{\lim }{\displaystyle {\int }_X(h_4\circ {\beta }_{k_{l_m}})\text{d}\mu =}{\displaystyle {\int }_X(h_4\circ \beta )\text{d}\mu }$(10)Thus,$\begin{array}{l}\left|\Phi (f_{k_{l_m}})-\Phi (f)\right|\\ =|{\displaystyle {\int }_X(h_1\circ {\alpha }_{k_{l_m}}-h_1\circ \alpha )\text{d}\mu +}{\displaystyle {\int }_X(h_3\circ {\beta }_{k_{l_m}}-h_3\circ \beta )\text{d}\mu }\\ +\text{i}({\displaystyle {\int }_X(h_2\circ {\alpha }_{k_{l_m}}-h_2\circ \alpha )\text{d}\mu +}{\displaystyle {\int }_X(h_4\circ {\beta }_{k_{l_m}}-h_4\circ \beta )\text{d}\mu })|\\ \le |{\displaystyle {\int }_X(h_1\circ {\alpha }_{k_{l_m}}-h_1\circ \alpha )\text{d}\mu }|+\left|{\displaystyle {\int }_X(h_3\circ {\beta }_{k_{l_m}}-h_3\circ \beta )\text{d}\mu }\right|\\ +|{\displaystyle {\int }_X(h_2\circ {\alpha }_{k_{l_m}}-h_2\circ \alpha )\text{d}\mu }|+\left|{\displaystyle {\int }_X(h_4\circ {\beta }_{k_{l_m}}-h_4\circ \beta )\text{d}\mu }\right|\end{array}$From (7)-(10), we conclude $\Phi (f_{k_{l_m}})\to \Phi (f)$. Hence, $\Phi$ is continuous.

Lemma 2  　 If the function $\Phi :L^p(C,\mu )\to C$ is a valuation, then$\Phi (f)=\Phi (\Re f)+\Phi (\text{i}\Im f)$for all $f\in L^p(C,\mu )$.

Proof  　If $\Re f,\Im f\ge 0$ or $\Re f,\Im f\le 0$, then, by (2) and (3), we have$\Phi (\Re f)+\Phi (\text{i}\Im f)=\Phi (f)+\Phi (0)$(11)If $\Re f\ge 0,\Im f\le 0$ or $\Re f\le 0,\Im f\ge 0$, then, by (2) and (3), we have$\Phi (f)+\Phi (0)=\Phi (\Re f)+\Phi (\text{i}\Im f)$(12)Note that $\Phi (0)=0$, apply (11) and (12), then get$\Phi (f)=\Phi (\Re f)+\Phi (\text{i}\Im f)$for all $f\in L^p(C,\mu )$.

If we restrict f to $L^p(R,\mu )$, then it is obvious that $\Phi$ is a valuation on $L^p(R,\mu )$. Also, we can construct another valuation on $L^p(R,\mu )$ which is related to $\Phi$.

Lemma 3  　Let $\Phi :L^p(C,\mu )\to C$ be a valuation. If the functions ${\Phi }^{\prime }:L^p(R,\mu )\to C$ is defined by${\Phi }^{\prime }(f)=\Phi (\text{i}f)$for all $f\in L^p(R,\mu )$, then ${\Phi }^{\prime }$ is a valuation on $L^p(R,\mu )$.

Proof  　For $f,g\in L^p(R,\mu )$, by (2) and (3), we have$\text{i}(f\vee g)=\text{i}f\vee \text{i}g\text{ and i}(f\wedge g)=\text{i}f\wedge \text{i}g$(13)By (13) and the valuation property of $\Phi$, it follows that$\begin{array}{l}{\Phi }^{\prime }(f\vee g)+{\Phi }^{\prime }(f\wedge g)\\ =\Phi (\text{i}f\vee \text{i}g)+\Phi (\text{i}f\wedge \text{i}g)\\ =\Phi (\text{i}f)+\Phi (\text{i}g)={\Phi }^{\prime }(f)+{\Phi }^{\prime }(g)\end{array}$for all $f,g\in L^p(R,\mu )$. Thus, ${\Phi }^{\prime }$ is a valuation on $L^p(R,\mu )$.

Lemma 4  　Let $\Phi :L^p(R,\mu )\to C$ be a valuation. If the functions ${\Phi }_1,{\Phi }_2:L^p(R,\mu )\to R$ are defined by$\Phi (f)={\Phi }_1(f)+\text{i}{\Phi }_2(f)$for all $f\in L^p(R,\mu )$, then both ${\Phi }_1,{\Phi }_2$ are real-valued valuations on $L^p(R,\mu )$.

Proof  　Since $\Phi$ is a valuation, we have$\begin{array}{l}\Phi (f\vee g)+\Phi (f\wedge g)\\ ={\Phi }_1(f\vee g)+\text{i}{\Phi }_2(f\vee g)\\ +{\Phi }_1(f\wedge g)+\text{i}{\Phi }_2(f\wedge g)\\ =\Phi (f)+\Phi (g)\\ ={\Phi }_1(f)+\text{i}{\Phi }_2(f)+{\Phi }_1(g)+\text{i}{\Phi }_2(g)\end{array}$for all $f,g\in L^p(R,\mu )$. Thus,${\Phi }_1(f\vee g)+{\Phi }_1(f\wedge g)={\Phi }_1(f)+{\Phi }_1(g)$and${\Phi }_2(f\vee g)+{\Phi }_2(f\wedge g)={\Phi }_2(f)+{\Phi }_2(g)$for all $f,g\in L^p(R,\mu )$.

Therefore, both ${\Phi }_1,{\Phi }_2$ are real-valued valuations on $L^p(R,\mu )$.

In order to establish a representation theorem for continuous complex-valued valuations on $L^p(C,\mu )$, we will use the corresponding representation theorem for real case which was obtained by Tsang ^[51].

Theorem 6  ^[51]　Let $(X,\aleph ,\mu )$ be a non-atomic measure space and let $\Phi :L^p(R,\mu )\to R$ be a continuous translation invariant valuation. If there exists a continuous function $h:R\to R$ with $h(0)=0$ such that $\Phi (b{\chi }_E)=h(b)\mu (E)$ for all $b\in R$ and all $E\in \aleph$ with $\mu (E)<\infty$, then there exist constants ${\gamma }_1,\delta \ge 0$ such that $\left|h(a)\right|\le \gamma {\left|a\right|}^p+\delta$ for all $a\in R$, and$\Phi (f)={\displaystyle {\int }_X\left(h\circ f\right)\text{d}\mu }$for all $f\in L^p(R,\mu )$. In addition, $\delta =0$ if $\mu (X)=\infty$.

Proof of Theorem 1  

Let $\Phi :L^p(C,\mu )\to C$ be a continuous valuation. For $f\in L^p(C,\mu )$, by Lemma 2, Lemma 3 and Lemma 4, we have$\begin{array}{l}\Phi (f)=\Phi (\Re f)+\Phi (\text{i}\Im f)\\ ={\Phi }_1(\Re f)+\text{i}{\Phi }_2(\Re f)+{\Phi }_1(\text{i}\Im f)+\text{i}{\Phi }_2(\text{i}\Im f)\\ ={\Phi }_1(\Re f)+\text{i}{\Phi }_2(\Re f)+{\Phi }_1{}^{\prime }(\Im f)+\text{i}{\Phi }_2^{\prime }(\Im f)\end{array}$where ${\Phi }_1^{\prime }(\Im f)={\Phi }_1(\text{i}\Im f)$ and ${\Phi }_2^{\prime }(\Im f)=\text{i}{\Phi }_2(\text{i}\Im f)$.

Since $\Re (f\vee g)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\Re f\vee \Re g,\Re (f\wedge g)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\Re f\wedge \Re g,$$\Im (f\vee g)=\Im f\vee \Im g,\text{\hspace{0.17em}and\hspace{0.17em}}\Im (f\wedge g)=\Im f\wedge \Im g$. Moreover, Lemma 3 and Lemma 4 imply that ${\Phi }_1,{\Phi }_2,{\Phi }_1^{\prime },{\Phi }_2^{\prime }$ are real-valued valuations on $L^p(R,\mu )$.

If we restrict to $f\in L^p(R,\mu )$, then the continuity of $\Phi$ implies that ${\Phi }_1,{\Phi }_2,$ are continuous on $L^p(R,\mu )$.

If we consider $f\in L^p(C,\mu )$ with $\Re f=0$, then the continuity of $\Phi$ implies that ${\Phi }_1^{\prime },{\Phi }_2^{\prime }$ are continuous on $L^p(R,\mu )$. Thus, ${\Phi }_1,{\Phi }_2,{\Phi }_1^{\prime },{\Phi }_2^{\prime }$ are continuous real-valued valuations on $L^p(R,\mu )$. It follows from Theorem 6 that there exist continuous functions $h_k:R\to R$ with the properties that $h_k(0)=0$ and there exist constants ${\gamma }_k,{\delta }_k\ge 0$ such that $\left|h_k(a)\right|\le$${\gamma }_k{\left|a\right|}^p+{\delta }_k$ for all $a\in R$(k = 1,2,3,4), and$\begin{array}{l}\Phi (f)={\displaystyle {\int }_X\left(h_1\circ \Re f+h_3\circ \Im f\right)\text{d}\mu }\\ +\text{\hspace{0.17em}i}{\displaystyle {\int }_X\left(h\circ \Re f+h_4\circ \Im f\right)\text{d}\mu }\end{array}$for all $f\in L^p(C,\mu )$. In addition, ${\delta }_k=0$(k = 1,2,3,4) if $\mu (X)=\infty$.

If µ is Lebesgue measure, then $L^p(C,\mu )$ becomes the space of Lebesgue measurable complex-valued functions. We usually write as $L^p(C,R^n)$.

Lemma 5  　Let $h_k:R\to R$ be continuous functions with the property that there exists ${\gamma }_k\ge 0$ such that $\left|h_k(a)\right|\le {\gamma }_k{\left|a\right|}^p$ for all $a\in R$(k = 1,...,4). If the function $\Phi :L^p(C,R^n)\to C$ is defined by$\begin{array}{l}\Phi (f)={\displaystyle {\int }_{R^n}\left(h_1\circ \Re f+h_3\circ \Im f\right)\text{d}x}\\ +\text{i}{\displaystyle {\int }_X\left(h_2\circ \Re f+h_4\circ \Im f\right)\text{d}x}\end{array}$Then $\Phi$ is a continuous translation invariant valuation.

Proof  　 Let M enote the collection of Lebesgue measurable sets in $R^n$. Take $X=R^n,\aleph =M$ and µ Lebesgue measure in Lemma 1 to conclude that $\Phi$ is a continuous valuation on $L^p(C,R^n)$.

For every $t\in R^n$ and every $f\in L^p(C,R^n)$, we have$\begin{array}{l}\Phi (f(x-t))={\displaystyle {\int }_{R^n}\left(h_1\circ \Re f+h_3\circ \Im f\right)(x-t)\text{d}x}\\ +\text{i}{\displaystyle {\int }_X\left(h_2\circ \Re f+h_4\circ \Im f\right)(x-t)\text{d}x}\\ =\Phi (f)\end{array}$which means that $\Phi$ is translation invariant.

Proof of Theorem 2  

It follows from Lemma 5 that (4) determines a continuous translation invariant valuation on $L^p(C,R^n)$. Conversely, let $\Phi :L^p(C,R^n)\to C$ be a continuous translation invariant valuation. Taking $X=R^n,$$\aleph =M$ and µ Lebesgue measure in the proof of Theorem 1, we obtain$\Phi (f)={\Phi }_1(\Re f)+\text{i}{\Phi }_2(\Re f)+{\Phi }_1^{\prime }(\Im f)+\text{i}{\Phi }_2^{\prime }(\Im f)$where ${\Phi }_1,{\Phi }_2,{\Phi }_1^{\prime },{\Phi }_2^{\prime }$ are real-valued valuations on $L^p(R,\mu )$. Theorem 4 implies that there exist continuous functions $h_k:R\to R$ with the property that there exist constants ${\gamma }_k\ge 0$ such that $\left|h_k(a)\right|\le {\gamma }_k{\left|a\right|}^p$ for all $a\in R$(k = 1,2,3,4), and$\begin{array}{l}\Phi (f)={\displaystyle {\int }_{R^n}\left(h_1\circ \Re f+h_3\circ \Im f\right)\text{d}x}\\ +\text{i}{\displaystyle {\int }_{R^n}\left(h_2\circ \Re f+h_4\circ \Im f\right)\text{d}x}\end{array}$for all $f\in L^p(C,R^n)$.

Let W denote the $\sigma$-algebra defined as$W=\left\{E:E\subseteq S^{n-1},\{\lambda x:x\in E,0\le \lambda \le 1\}\in M\right\}$

Also denote by $\sigma$ the spherical Lebesgue measure. If µ is the spherical Lebesgue measure, then $L^p(C,\sigma )$ denotes the space of spherical Lebesgue measurable complex valued functions. We usually write as $L^p(C,S^{n-1})$.