Yu. I. Gilderman

ABSTRACT SET FUNCTIONS AND S. L. SOBOLEV EMBEDDING THEOREMS

(Presented by Academician S. L. Sobolev on 23 I 1962)

In this paper we consider additive abstract set functions \(\varphi(E)\), defined for all \(L\)-measurable sets \(E\) from some \(n\)-dimensional domain \(\Omega \in R_n\), with values in a \(B\)-space \(X\). A detailed study of such functions was carried out in \((^{1,7})\). Some results are also contained in \((^{3,4})\).

Most of the concepts and notation that we use were introduced in \((^{1,3})\). Only occasionally, instead of \(\Psi_p\) or \(\Phi_p\), following V. B. Korotkov, shall we write \(\Psi_p(\Omega)\) or \(\Psi_p(X,\Omega)\) (similarly \(\Phi_p(\Omega)\) and \(\Phi_p(X,\Omega)\)).

We shall consider abstract additive functions belonging to the \(B\)-space \(\Phi_1(\Omega)\) with norm \(\sup\limits_{E_1 \cap E_2=0}\|\varphi(E_1)-\varphi(E_2)\|_X\), or with the equivalent norm \(\sup\limits_{E \subset \Omega}\|\varphi(E)\|_X\), as well as functions from the \(B\)-spaces \(\overline{\Phi}_1(\Omega)\), \(\overline{\overline{\Phi}}_1(\Omega)\), considered in \((^3)\). Here \(\overline{\overline{\Phi}}_1\) is the space of absolutely continuous functions, i.e., such functions for which \(\|\varphi(E)\|_X \to 0\) as \(mE \to 0\), and \(\overline{\Phi}_1\) is the space of additive and normal functions, i.e., additive functions tending to \(0\) in the norm of \(X\) on a vanishing sequence of sets.

It is easy to establish \((^3)\) that \(\overline{\Phi}_1\) coincides with the space of countably additive abstract functions. In addition, we shall consider the space \(\Phi_p(\Omega)\), the space of additive functions with bounded norm

\[ \|\varphi\|_{\Phi_p} = \sup_{\omega \in L_{p'}} \frac{ \left\|\int_\Omega \omega\, d\varphi(E)\right\|_X }{ \|\omega\|_{L_{p'}} }, \qquad p>1,\quad \frac{1}{p}+\frac{1}{p'}=1. \]

It can be shown \((^{1,3})\) that

\[ \Phi_p \subset \overline{\overline{\Phi}}_1 \subset \overline{\Phi}_1 \subset \Phi_1. \]

Let \(\varphi(E)\in \overline{\Phi}_1(X,\Omega)\).

Theorem 1. If \(\varphi(E)\) is normal, then \(\psi_\varphi(E')=\varphi(E'\cap E)\) is also normal as an abstract function of sets \(E'\) with values in \(\overline{\Phi}_1(X,\Omega)\).

In other words, if \(\varphi(E)\in \overline{\Phi}_1\) and \(E_k\) is a vanishing sequence, then
\[ \sup_{E\in\Omega}\|\varphi(E\cdot E_k)\|_X = \|\varphi\|_{\Phi_1(E_k)} \to 0 \quad \text{as } k\to\infty . \]

The theorem follows easily from the equality
\[ E\cap\left[\bigcup_i E'_i\right] = \bigcup_i [E\cap E'_i], \]
where the \(E'_i\) are disjoint, and from theorem 3 \((^3)\).

From theorem 1, in particular, there follows a very simple proof of the following theorem (see \((^3)\)).

Theorem 2. If \(\varphi(E)\in\overline{\Phi}_1\), \(E_{k+1}\subset E_k\), and \(E_0=\lim\limits_{k\to\infty}E_k\), while \(\omega(x)\) is a bounded numerical function such that \(\omega(x)=a_0\) for \(x\in E_0\), then

\[ \lim_{k\to\infty}\int_{E_k}\omega(x)\,d\varphi(E) = a_0\varphi(E_0). \]

Indeed:

\[ \left\| \int_{E_k} \omega(x)\, d\varphi(E) - a_0\varphi(E_0) \right\|_X = \]

\[ = \left\| \int_{E_k-E_0} \omega(x)\, d\varphi(E) \right\|_X \leq \max_{x\in\Omega} |\omega(x)| \, \|\varphi\|_{\Phi_1(E_k-E_0)} \leq \varepsilon \]

for \(k>K(\varepsilon)\), since \(\{E_k-E_0\}\) is a vanishing sequence of sets.

Theorem 3. If \(E_k\in\Omega\) is an increasing (decreasing) sequence of sets, \(E_0=\lim_{k\to\infty} E_k\), and \(\varphi(E)\in\overline{\Phi}_1\), then

\[ \lim_{k\to\infty}\|\varphi(E_k)-\varphi(E_0)\|_X=0, \]

\[ \lim_{k\to\infty}\|\varphi\|_{\Phi_1(E_k)}=\|\varphi\|_{\Phi_1(E_0)}. \]

Theorem 4. Let \(E_n\) be any sequence of sets from \(\Omega\). Denote

\[ E_0^s=\limsup_n E_n=\bigcap_{k=1}^{\infty}\bigcup_{n=k}^{\infty} E_n \quad\text{and}\quad E_0^i=\liminf_n E_n=\bigcup_{k=1}^{\infty}\bigcap_{n=k}^{\infty} E_n. \]

Then, if \(\varphi(E)\in\overline{\Phi}_1\), then

\[ \|\varphi\|_{\Phi_1(E_0^s)}\geq \overline{\lim_n}\,\|\varphi\|_{\Phi_1(E_n)}, \]

\[ \|\varphi\|_{\Phi_1(E_0^i)}\leq \underline{\lim_n}\,\|\varphi\|_{\Phi_1(E_n)}. \]

From Theorem 4 it follows:

Theorem 5. In order that a normal function be absolutely continuous, it is necessary and sufficient that it vanish on every set of measure zero.

This theorem generalizes the well-known theorem for numerical normal set functions (see, for example, \((^5)\), p. 53).

Let now \(\varphi(E)\in\Phi_p\), \(p>1\), or \(\in\overline{\Phi}_1\) for \(p=1\).

Lemma 1. If \(\varphi(E)\in\Phi_p\), \(p>1\) \((\varphi(E)\in\overline{\Phi}_1\) for \(p=1)\), \(\omega_1(x)\in L(\Omega)\), \(\omega_2(x)\in L_{p'}(\Omega)\), \(\omega_1=\omega_2=0\) outside \(\Omega\), \(\frac{1}{p}+\frac{1}{p'}=1\), then in the triple convolution

\[ \psi(y)=\int \omega_1(z)\left[\int \omega_2(y-z-x)\,d_x\varphi(E)\right]dz \tag{1} \]

the associativity law is valid and the order of performing the operations may be changed. The formula holds

\[ \psi(y)=\int\left[\int \omega_1(z)\omega_2(y-z-x)\,dz\right]d_x\varphi(E). \]

This lemma for bounded \(\omega_1(x)\) and \(\omega_2(x)\) and for \(\varphi(E)\in\Psi_p(\Omega)\), \(p>1\), was proved in (1) (\(\Psi_p\) is the space of functions from \(\Phi_p\) continuous with respect to translation in the norm \(\Phi_p\)).

From Lemma 1 and Lemmas III and IV of (1), pp. 314–315, follows the equality

\[ [z_\varphi(y)]_h=z_{\varphi_h}(y), \]

where \(z_\varphi(y)=\varphi(E+y)\).

Hence it is easily obtained:

Theorem 6. If \(\varphi(E)\in\Phi_p\) (\(\overline{\Phi}_1\) for \(p=1\)) and \(z_\varphi(y)\) is measurable as a function of the point \(y\) with values in \(\Phi_p\) \((\overline{\Phi}_1)\), then \(\varphi(E)\) is continuous with respect to translation in the norm \(\Phi_p\) \((\overline{\Phi}_1)\).

A certain modification of Lemma 1 is

Lemma 2. If \(\varphi(E)\in\overline{\Phi}_1\) and is continuous with respect to translation in the norm \(\Phi_1\), and \(\omega_1(x)\) and \(\omega_2(x)\) are bounded functions, then in the triple convolution (1) the associativity law is also valid.

Hence it is also easy to obtain the equality \(z_{\varphi_h}(y)=[z_\varphi(y)]_h\), whence it follows:

Theorem 7. The mean functions \(\varphi_h(E)\) are dense in the space \(\Psi_1(\Omega)\), where \(\Psi_1(\Omega)\) is the space of normal functions continuous under translation in the metric \(\Phi_1(\Omega)\).

From this it follows immediately:

Theorem 8. If a function \(\varphi(E)\in\Psi_1(\Omega)\), then it is absolutely continuous in the metric \(\Phi_1(\Omega)\).

This theorem generalizes a similar theorem for numerical functions (see, for example, \((^5)\), p. 140).

Let us now consider the space \(\Phi_{\mathbf p}^{\,l}(\Omega)\), \(\mathbf p=(p_1,p_2,\ldots,p_n)\), \(p_i\geqslant 1\), \(l=(l_1,l_2,\ldots,l_n)\), \(l_i\geqslant 1\). To the space \(\Phi_{\mathbf p}^{\,l}(\Omega)\) we shall assign all normal functions \(\varphi(E)\) whose generalized derivatives (see (1)) \(\psi^{l_i}(E)\) of order \(l_i\) in the direction \(x_i\) belong to \(\Phi_{p_i}(\Omega)\) \((\overline{\Phi}_1(\Omega),\ p_i=1)\).

Theorem 9. If a function \(\varphi(E)\) belongs to \(\Phi_{\mathbf p}^{\,l}(\Omega)\), then it is constant under translation on sets \(E\in\Omega\) whose measure is zero.

From this it follows:

Theorem 10. A function \(\varphi(E)\in\Phi_{\mathbf p}^{\,l}\) is equal to zero on a set of zero measure \(u\), and consequently is absolutely continuous.

Obviously, \(\Psi_{\mathbf p}^{\,l}\subset\Phi_{\mathbf p}^{\,l}(\Omega)\), where \(\Psi_{\mathbf p}^{\,l}\) is the space of normal functions having all generalized derivatives \(\Psi^1(E)\in\Phi_{\mathbf p}\), \(|1|=l\). From Theorem 10 it follows that for the derivative function \(\varphi(E)\in\Phi_{\mathbf p}^{\,l}\) the equality

\[ \psi^1(E)=\lim_{|\Delta x_i|\to 0} \frac{\Delta_{x_1}^{l_1}\cdots \Delta_{x_n}^{l_n}\varphi(E)} {|\overrightarrow{\Delta x_i}|^l}, \qquad l=|\vec{1}|=l_1+l_2+\cdots+l_n, \]

holds, where \(\Delta_{x_1}^{l_1}\cdots \Delta_{x_n}^{l_n}\varphi(E)\) denotes the corresponding \(l\)-th difference of the function \(\varphi(E)\), and the limit is understood in the sense of the metric \(\Phi_1(\Omega)\).

The last equality can also be rewritten in the form

\[ z_{\psi^1}(x)= \frac{\partial^l z_\varphi(x)} {\partial x_1^{l_1}\cdots \partial x_n^{l_n}}, \]

where \(z_{\psi^1}(x)\) and \(z_\varphi(x)\) are abstract functions of points with values in \(\Phi_1(X,\Omega)\). Hence, for \(\Psi^1(E)\in\Psi_{\mathbf p}\) for all \(|1|=l\), Theorem 6 \((^3)\) follows at once.

Let now \(X=R_1\). From Theorem 8 and the Radon–Nikodym theorem (see, for example, \((^5)\), p. 59) it follows:

Theorem 11. The space of numerical functions \(W_p^l(\Omega)\), introduced by S. L. Sobolev \((^2)\), coincides with the space of normal additive numerical functions of a set having all generalized derivatives of order \(l\) belonging to the space \(\Phi_p(R_1,\Omega)=L_p(\Omega)\) \((p>1)\).

In conclusion, I express my gratitude to Academician S. L. Sobolev for his attention to the work and to V. B. Korotkov for useful discussions.

Institute of Mathematics with Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR

Received
17 I 1962

REFERENCES CITED

\(^1\) S. L. Sobolev, Fund. math., 47, 3, 277 (1959).
\(^2\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\(^3\) Yu. I. Gilderman, DAN, 140, No. 4 (1961).
\(^4\) V. B. Korotkov, DAN, 141, No. 2 (1961).
\(^5\) S. Saks, Theory of the Integral, M., 1949.
\(^6\) V. I. Smirnov, Course of Higher Mathematics, 5, M., 1959.
\(^7\) N. Dunford, I. T. Schwartz, Linear Operators, part I, N. Y., 1958.