Mathematics

B. P. Kufarev, N. G. Nikulina

Lebesgue Measure of Subsets of Euclidean Space as the Highest Variation of the Distance Function to a Closed Set*

(Presented by Academician M. A. Lavrent'ev on 24 VIII 1964)

Let \(G\) be a bounded open set in the Euclidean plane \(R_2\). At each point \(z \in R_2\) with coordinates \(x, y\), the function
\(r(x,y)=r(z)=r(z,R_2\setminus G)\) is defined, where \(r\) is distance in \(R_2\). The set
\(E_t=\{z: r(z)=t\}\) is called the \(t\)-level of the function \(r(z)\).

Theorem 1. For almost all \(t \in (0,T]\), where \(T=\max\limits_{z\in G} r(z)\), each component \(E_t^\alpha\) of the set \(E_t\) is a closed rectifiable Jordan curve (whose length \(l_t^\alpha\) is in the usual sense) or a point; the length \(l_t\) of the set \(E_t\) (equal to \(\sum_\alpha l_t^\alpha\)) as a function of \(t\) is summable on the interval \((0,T]\), and the planar Lebesgue measure

\[ \operatorname{mes}_2 G=\int_0^T l_t\,dt. \tag{1} \]

Proof. Since \(|r(z')-r(z'')|\le r(z',z'')\) for any points \(z',z''\in R_2\) ((\(^{3}\), p. 373), the function \(r(z)\) satisfies a Lipschitz condition with constant 1. By a known theorem of V. V. Stepanov ((\(^{4}\), p. 449), at almost every point the function \(r(z)\) has a total differential. Obviously, at every such point \(z_1\in G\),
\(|\operatorname{grad} r(z_1)|\le 1\).

Let \(z\in R_2\setminus G\) be such that \(r(z_1)=r(z_1,z)\), and let \(z_2\) lie on the segment \(\overline{z_1z}\). It is easy to show that
\(r(z_1)-r(z_2)=r(z_1,z_2)\). Therefore for the function \(-r(z)\) the direction of the gradient coincides with the direction of the vector \(\overrightarrow{z_1z}\), whence it follows that almost everywhere in \(G\)

\[ |\operatorname{grad} r(z)|=|\operatorname{grad}[-r(z)]|=1. \tag{2} \]

The Steklov means

\[ r_h(z)=r_h(x,y)=\frac{1}{h^2}\int_0^h\int_0^h r(x+\xi,y+\eta)\,d\xi\,d\eta \]

of the function \(r(z)\), absolutely continuous in each of the arguments \(x,y\), have the following well-known properties:

1) at every point \(z\in R_2\), the function \(r_h(x,y)\) has continuous partial derivatives with respect to \(x\) and to \(y\);

2) \(\partial r_h/\partial \sigma=[\partial r/\partial \sigma]_h\) at points of differentiability of \(r(z)\) (here \([\partial r/\partial \sigma]_h\) is the Steklov mean of the derivative \(\partial r/\partial \sigma\); \(\sigma=x,y\));

* The present note is adjacent to the works (\(^{1,2}\)); we use some concepts and theorems from these works.

3) as \(h\to 0\), \(r_h(z)\to r(z)\) uniformly on every bounded closed set \(F\subset R_2\).

Applying the Cauchy–Bunyakovsky inequality, we obtain, for every point \(z\in R_2\) at which \(r(z)\) is differentiable,

\[ |\operatorname{grad} r_h(z)|^2 = \left[\frac{\partial r}{\partial x}\right]_h^2 + \left[\frac{\partial r}{\partial y}\right]_h^2 \le \frac{1}{h^2} \int_0^h\int_0^h |\operatorname{grad} r(x+\xi,y+\eta)|^2\,d\xi\,d\eta \le 1, \]

i.e. \(|\operatorname{grad} r_h(z)|\le 1\) almost everywhere in \(R_2\). By property 1), the last inequality is valid everywhere in \(R_2\).

Therefore, using Theorem 33 of \((^1)\), for the planar variation \(W\) of the function \(r_h(z)\) we have:

\[ W(r_h,I) = \int_{-\infty}^{\infty} v(E_t^h)\,dt = \iint_I |\operatorname{grad} r_h(x,y)|\,dx\,dy \le S(I), \]

where \(S(I)\) is the area of the fixed square \(I\subset R_2\); \(v(E_t^h)\) is the linear Hausdorff measure \(((^4),\ \text{p. }86)\) of the \(t\)-level \(E_t^h\) of the function \(r_h(z)\). Further, on every square \(I\subset R_2\) the planar variation of the function \(r(z)\) is bounded, since it is a lower semicontinuous functional with respect to uniform convergence \(((^1),\ \text{p. }90)\), and

\[ W(r,I)\le \lim_{h\to 0} W(r_h,I)\le S(I). \]

It follows that the function \(r(z)\) is absolutely continuous (in the sense of Kronrod). Indeed, let \(\varepsilon>0\) be arbitrary and let \(N\) be a bounded Borel set with measure less than \(\delta=\varepsilon\). Cover \(N\) by a countable system of nonoverlapping squares \(\{I_i\}\), \(i=1,2,\ldots\), with sum of areas \(<\delta\). Then, by the complete additivity of planar variation on the class of Borel sets,

\[ W(r,N)\le W\left(r,\bigcup_i I_i\right) \le \sum_i W(r,I_i) \le \sum_i S(I_i)<\varepsilon. \]

By Theorem 32 of \((^1)\), for a square \(I\supset \overline{G}\) (\(\overline{G}\) is the closure of \(G\)) we have:

\[ W(r,I) = \int_0^T v(E_t)\,dt = \iint_I |\operatorname{grad} r(z)|\,dx\,dy, \tag{3} \]

whence, by virtue of equality (2),

\[ \operatorname{mes}_2 G \le \int_0^T v(E_t)\,dt \le \operatorname{mes}_2 \overline{G}. \tag{3'} \]

Let \(\tau\in(0,T]\), \(G^\tau=\{z:r(z,R_2\setminus G)>\tau\}\), and \(r_\tau(z)=r(z,R_2\setminus G^\tau)\). It is easy to see that the \(t\)-level \(E_t^\tau\) of the function \(r_\tau(z)\) coincides, for \(t\ge \tau\), with the \((t+\tau)\)-level of the function \(r(z)\).

Applying inequality \((3')\) to the function \(r_\tau(z)\), we have

\[ \operatorname{mes}_2 \overline{G}^{\,\tau} \ge \int_0^{T-\tau} v(E_t^\tau)\,dt = \int_\tau^T v(E_t)\,dt, \]

and, since \(\overline{G}^{\,\tau}\subset G\), therefore

\[ \operatorname{mes}_2 G \ge \int_0^T v(E_t)\,dt, \]

which together with \((3')\) gives

\[ \operatorname{mes}_2 G = \int_0^T v(E_t)\,dt. \tag{4} \]

Since \(v(E_t)\) is finite for almost all \(t\) (see (3)), Theorem 10 of \((^1)\) permits us to assert that, for almost all \(t\in(0,T]\), the \(t\)-level of the function

\(r(z)\) consists of point components or entirely regular components, with
\(\nu(E_t)=\sum_\alpha \nu(E_t^\alpha)<\infty\). For such \(t\) the component \(E_t^\alpha\) is locally connected ((1), p. 51) and contains no branch points; therefore \(E_t^\alpha\) is homeomorphic to a circle ((1), p. 62), and \(\nu(E_t^\alpha)=l_t^\alpha\) ((4), p. 190), which completes the proof of the theorem.

Remark 1. If the boundary \(\partial G\) of the set \(G\) has nonzero plane Lebesgue measure, then from (3) and (4) it follows that \(\operatorname{grad} r(z)=0\) almost everywhere on \(\partial G\).

Remark 2. For the case where \(G\) is a simply connected domain, in note (5) the supposition was made that the level set \(E_t\) has finite length for all \(t\). Theorem 1 justifies this supposition for almost all \(t \in (0,T]\), even when \(G\) is an arbitrary bounded open set. We also note that in the case considered by us, as in (5), a metric characterization of all \(t\)-levels is possible; namely, assertions analogous to assertions I—IV of the theorem cited in (5) are valid.

Quite analogously to Theorem 1 one proves

Theorem 2. Let \(G\) be a bounded open set in Euclidean space \(R_n\), \(x \in R_n\) an arbitrary point, and \(r(x)=r(x,R_n\setminus G)\), where \(r\) is the distance in \(R_n\). Then the \((n-1)\)-dimensional Hausdorff measure \(\nu(E_t)\) of the set \(E_t=\{x:r(x)=t\}\), as a function of \(t\), is summable on the interval \((0,T]\), \(T=\max_{x\in G} r(x)\), and moreover the \(n\)-dimensional Lebesgue measure is

\[ \operatorname{mes}_n G=\int_0^T \nu(E_t)\,dt . \tag{5} \]

From formula (5) it follows that the \(t\)-level \(E_t\) has finite \((n-1)\)-dimensional Hausdorff measure for almost all \(t\in(0,T]\); however, apparently, it can be proved that \(\nu(E_t)<\infty\) for all \(t\).

Theorem 2 admits the following generalization.

Theorem 3. Let \(M\subset R_n\) be an arbitrary measurable set, and let \(F\subset R_n\) be such a closed set that \(\operatorname{mes}_n(F\cap M)=0\), and let \(E_t^F\) be the \(t\)-level of the function \(r_F(x)=r(x,F)\), where \(r\) is the distance in \(R_n\) \((x\in R_n)\).

Then the \((n-1)\)-dimensional Hausdorff measure \(\nu(E_t^F\cap M)\), as a function of \(t\), is summable on the interval \((0,\infty)\), and moreover

\[ \operatorname{mes}_n M=\int_0^\infty \nu(E_t^F\cap M)\,dt . \tag{6} \]

It is evident that formula (6) generalizes the known formula relating the measure of a set in \(R_n\) to the measures of its sections (see (3), p. 371, or (6), p. 143). In the case \(n=2\), for almost all \(t\), the Hausdorff measure \(\nu(E_t^F\cap M)\) can be replaced by the Lebesgue measure \(m_t(E_t^F\cap M)\) on the curve \(E_t^F\).

Tomsk State University
named after V. V. Kuibyshev

Received
17 VIII 1964

CITED LITERATURE

\(^{1}\) A. S. Kronrod, Uspekhi Mat. Nauk, 5, 1(35), 24 (1950).
\(^{2}\) A. G. Vitushkin, On Multidimensional Variations, 1956.
\(^{3}\) I. P. Natanson, Theory of Functions of a Real Variable, 1957.
\(^{4}\) S. Saks, Theory of the Integral, 1949.
\(^{5}\) V. K. Ionin, G. D. Suvorov, Dokl. Akad. Nauk SSSR, 129, No. 3 (1959).
\(^{6}\) P. R. Halmos, Measure Theory, 1953.