UDC 517.5

MATHEMATICS

Yu. I. GILDERMAN

ON ADDITIVE FUNCTIONS OF SETS CONTINUOUS IN A DIRECTION

(Presented by Academician S. L. Sobolev on 10 III 1966)

We shall consider countably additive real functions of sets \(\Phi(E)\), defined on Borel sets \(E\) of the \(n\)-dimensional cube \(\Omega\) with sides parallel to the coordinate axes \(x_1, x_2, \ldots, x_n\). For an arbitrary \(E\) from \(n\)-dimensional Euclidean space \(R_n\) we shall put \(\Phi(E)=\Phi(E\cap\Omega)\). By \(W\Phi(E)\) we shall denote the total variation of the function \(\Phi\) on \(E\). Finally, together with the function \(\Phi(E)\), we shall consider the function \(\Phi^h(E)=\Phi(E+h)\), where \(h\) is a vector in \(R_n\), and \(E+h\) denotes the set \(E\) shifted as a rigid body by the vector \(h\).

Classical results (see, for example, \((^1)\)) assert the equivalence of the following three conditions:

\(1^\circ.\) The function \(\Phi(E)\) is absolutely continuous with respect to Lebesgue measure.

\(2^\circ.\)
\[ \lim_{|h|\to 0} W[\Phi^h-\Phi](\Omega)=0 \tag{1} \]
(the function \(\Phi(E)\) is continuous with respect to translation).

\(3^\circ.\)
\[ \Phi(E)=\int_E f(x)\,dx,\qquad x\in R_n. \tag{2} \]

The present note is devoted to countably additive functions possessing properties \(1^\circ\) or \(2^\circ\) only along certain coordinate axes. In this case, as we shall show, conditions \(1^\circ\) and \(2^\circ\) cease to be equivalent. Functions possessing these properties are, generally speaking, singular. However, for them one can obtain certain integral representations analogous to (2).

Let \(m_k A\) denote the \(k\)-dimensional Lebesgue measure of the set \(A\), and let \(P_i A\) be the projection of the set \(A\) onto the axis \(x_i\).

Definition 1. The function \(\Phi(E)\) is called absolutely continuous in the coordinate direction \(x_i\) if \(|\Phi(E)|\to 0\) as soon as \(m_1P_iE\to 0\).

Example 1. The function of plane sets
\[ \varphi(E)=m_1P_1(E\cap d)=m_1P_2(E\cap d), \]
where \(d\) is the diagonal of the unit square, is an example of a function absolutely continuous in each coordinate direction.

The function \(\varphi(E)\) is obviously singular. Thus, absolute continuity in each coordinate direction is insufficient for ordinary absolute continuity.

Let \(P_{R_s}E\) denote the projection of the set \(E\) onto the subspace \(R_s,\ s<n\).

Definition 2. The function \(\Phi(E)\) is called absolutely continuous with respect to the collection of coordinate directions \(x_1, x_2, \ldots, x_s,\ s<n\), if \(|\Phi(E)|\to 0\) as soon as \(m_sP_{R_s}E\to 0\).

From absolute continuity with respect to the collection \(x_1,x_2,\ldots,x_s\) it obviously follows that there is absolute continuity in each of the directions \(x_1,x_2,\ldots,x_s\). The converse is not true.

Let \(R_s^{\perp}\) denote the orthogonal complement of \(R_s\), \(s<n\), in \(R_n\), and let \(BG\) be the totality of all Borel sets from some set \(G\subset R_n\).

Theorem 1. In order that the function \(\Phi(E)\) be absolutely continuous with respect to the totality of directions \(x_1,x_2,\ldots,x_s\), \(s<n\), it is necessary and sufficient that one of the following conditions be fulfilled:

1) \(\Phi(E)=0\), if \(m_sP_{R_s}E=0\).

2) For sets \(E\) of the form \(E=E_s\times E_s^{\perp}\), where \(E_s\in BR_s\), and \(E_s^{\perp}\in BR_s^{\perp}\), the representation holds
\[ \Phi(E_s\times E_s^{\perp})=\int_{E_s} f(E_s^{\perp},x^s)\,dx^s, \]
where \(x^s\in R_s\); \(f(E_s^{\perp},x^s)\), for almost all \(x^s\), is countably additive in \(E_s^{\perp}\) and, for all \(E_s^{\perp}\in BR_s^{\perp}\), is summable in \(x^s\).

It follows from 1) that from the set of singularities of a function \(\Phi(E)\) absolutely continuous with respect to the totality of directions \(x_1,x_2,\ldots,x_s\), one may remove any set \(Q\) such that \(m_sP_{R_s}Q=0\), without this set ceasing to be a set of singularities.

Theorem 2. If \(\Phi(E)\) is absolutely continuous with respect to the totality of directions \(x_1,x_2,\ldots,x_s\), then the functions \(\overline{W}\Phi(E)\), \(\underline{W}\Phi(E)\), and \(W\Phi(E)\) will have the same property. For \(W\Phi(E)\) the representation holds
\[ W\Phi(E_s\times E_s^{\perp})=\int_{E_s} Wf(E_s^{\perp},x^s)\,dx^s. \]

Analogous representations hold for the upper and lower variations \(\overline{W}\Phi(E)\) and \(\underline{W}\Phi(E)\).

Let \(x_s^{\perp}\in R_s^{\perp}\). Denote \(\{x_s^{\perp}\}=x_s^{\perp}\times R_s\).

Definition 3. The function \(\Phi(E)\) is called continuous under shifts with respect to the totality of directions \(x_1,x_2,\ldots,x_s\), if \(|\Phi(E)|\to0\) as soon as \(m_s[\{x_s^{\perp}\}\cap E]\to0\) simultaneously for all \(\{x_s^{\perp}\}\).

Lemma. Let \(Q\) be a set from \(B\Omega\) such that for any \(\{x_s^{\perp}\}\) the measure \(m_s[Q\cap\{x_s^{\perp}\}]=0\). Then for almost all, in the sense of the measure \(m_s\), vectors \(h\in R_s\), the equality
\[ \Phi^h(Q)=\Phi(Q+h)=0 \]
is valid.

Theorem 3. In order that the function \(\Phi(E)\) be continuous under shifts with respect to the totality of directions \(x_1,x_2,\ldots,x_s\), \(s<n\), it is necessary and sufficient that one of the following conditions be fulfilled:

1. The function \(\Phi(E)\) is absolutely continuous with respect to the measure \(\psi(E)\), defined by the equality
   \[ \psi(E_s\times E_s^{\perp})=W\Phi(\Omega_s\times E_s^{\perp})\cdot m_sE_s, \]
   where \(\Omega_s=P_{R_s}\Omega\), and, consequently,
   \[ \Phi(E)=\int_E f(x)\,d\psi(E). \]

2. The function \(\Phi(E)\) satisfies condition (1) only for vectors \(h\in R_s\).

3. The function \(\Phi(E)\) vanishes on any set \(E\in BR_n\) such that \(m_s[E\cap\{x_s^{\perp}\}]=0\) for any \(\{x_s^{\perp}\}\).

Thus, any of the properties 1, 2, and 3 listed in Theorem 3 may be taken as the definition of a function continuous under shifts.

Corollary 1. If \(\Phi(E)\) is continuous under shifts with respect to the totality \(x_1,x_2,\ldots,x_s\), then it is absolutely continuous with respect to this totality.

The converse, as we have seen, is false.

Corollary 2. The projection onto \(R_s^\perp\) of the set of singularities of a function \(\Phi(E)\) that is continuous under shifts jointly in \(x_1, x_2, \ldots, x_s\) has \((n-s)\)-dimensional measure zero.

Corollary 3. If the function \(\Phi(E)\) is continuous under shifts jointly in \(x_1, x_2, \ldots, x_s\) and absolutely continuous jointly in \(x_{s+1}, x_{s+2}, \ldots, x_n\), then it is absolutely continuous in the usual sense.

Example 2. The function \(\eta(E)\), defined for sets \(E = E_s \times E_s^\perp\) by the equality
\[ \eta(E)=\gamma(E_s)\cdot \delta(E_s^\perp), \]
is continuous under shifts jointly in \(x_1, x_2, \ldots, x_s\), if \(\gamma(E_s)\) is an absolutely continuous set function, \(E_s \in BR_s\), and \(\delta(E_s^\perp)\) is a countably additive set function, \(E_s^\perp \in BR_s^\perp\).

The following example shows that the simultaneous fulfillment of Corollaries 1 and 2 is only a necessary condition for shift continuity of the function \(\Phi(E)\).

Example 3. In the plane square \(\Omega=\{0\le x\le 1,\ 0\le y\le 1\}\) construct the graph of the “Cantor staircase” \(y=f(x)\) and remove from this graph the intervals on which the function \(f(x)\) is constant. Denote by \(C\) the remaining set of points on the graph, and define a set function \(E\subset\Omega\) by the formula
\[ \varphi(E)=m_1P_y(E\cap C), \]
where \(P_y\) denotes projection onto the \(y\)-axis. The function thus constructed is, obviously, absolutely continuous in the \(y\)-direction, since \(\varphi(E)=0\) if \(m_1P_yE=0\). The projection of its set of singularities (the set \(C\)) onto the \(x\)-axis has measure zero. At the same time, \(\varphi(E)\) is not continuous under shifts along \(y\).

The properties of set functions considered here are connected with differentiation in the generalized sense of S. L. Sobolev (see (2), and also (3)). We indicate two simple propositions:

1. If a countably additive set function \(\Phi(E)\) has, as generalized derivatives in the directions \(x_1, x_2, \ldots, x_s\), also countably additive functions, then \(\Phi(E)\) is continuous under shifts jointly in \(x_1, x_2, \ldots, x_s\).

2. If a countably additive function \(\Phi(E)\) is absolutely continuous jointly in \(x_2, x_3, \ldots, x_n\), then as its primitive in the direction \(x_1\) one may take a set function that is absolutely continuous in the usual sense.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
1 III 1966

REFERENCES

1. S. Saks, Theory of the Integral, IL, 1949.
2. S. L. Sobolev, Fund. Math., 47, No. 3, 277 (1959).
3. Yu. I. Gil'derman, Sibirsk. Mat. Zh., 6, No. 4, 727 (1965).