UDC 519.53

MATHEMATICS

A. G. VITUSHKIN

PROOF OF THE UPPER SEMICONTINUITY OF THE VARIATION OF A SET

(Presented by Academician A. N. Kolmogorov on 15 VI 1965)

In this note it is proved that the \(k\)-dimensional variation \(V_k(e)\) of a closed set \(e\), situated in Euclidean space, is upper semicontinuous if the variations of smaller dimension are bounded. Let us recall the definition of \(k\)-dimensional variation (see \((^1)\)).

Let \(E_n\) be \(n\)-dimensional Euclidean space; \(\Omega_n^k\) the space of \((n-k)\)-dimensional planes \(\beta_{n-k}\) in \(E_n\); \(\mu_n^k\) the Haar measure in the space \(\Omega_n^k\), invariant with respect to the group of transformations of the space \(\Omega_n^k\) generated by motions of the space \(E_n\); \(V_0(e)\) the number of components of the set \(e\). The \(k\)-th variation of a closed set \(e\) is the number

\[ V_k(e)=\int_{\Omega_n^k} V_0(e\cap \beta_{n-k})\,d\mu_n^k . \]

The normalization of the measure \(\mu_n^k\) is chosen so that the \(k\)-th variation of a \(k\)-dimensional unit cube is equal to 1.

In the space of subsets of \(E_n\) we shall use the deviation metric, i.e., as the distance between sets \(e\) and \(f\) from \(E_n\) we take the quantity
\(r(e,f)=\max[\rho(e,f),\rho(f,e)]\), where
\(\rho(A,B)=\sup_{x\in A}\rho(x,B)\), and \(\rho(x,B)\) is the distance from the point \(x\) to the set \(B\).

After these explanations we formulate precisely our result.

Theorem 1. If, for closed sets \(e\) and \(e_1,e_2,\ldots,e_i,\ldots\),

1) \(\displaystyle \lim_{i\to\infty} r(e,e_i)=0\);

2) all \(V_m(e_i)\) \((m=0,1,2,\ldots,(k-1);\ i=1,2,\ldots)\) do not exceed some constant \(M\),

then

\[ \sup_{i\to\infty} V_k(e_i)\geq V_k(e). \]

Consider the minimal \(\sigma\)-algebra \(\nu_{k,n}\) of sets in \(E_n\) containing all closed subsets of \(E_n\) with finite variations whose dimension is not greater than \(k\).

The variation \(V_k(e)\), extended to all sets \(e\in \nu_{k,n}\), turns out to be a countably additive measure of order \(k\), which is semicontinuous (see the formulation of the theorem) and on all polyhedra coincides with \(k\)-dimensional Lebesgue measure. It seems plausible that a measure satisfying the listed conditions is unique, i.e., coincides with \(V_k(e)\).

Notation.

\[ d(g)=\sup_{a\in g,\ b\in g}\rho(a,b) \]
is the diameter of the set \(g\); \(\partial(e)\) is the boundary of the set \(e\); the depth of immersion of the set \(e\) in \(g\cap\beta\) will be called
\[ r(e,g,\beta)=\sup \rho(x,\partial(g\cap\beta)), \]
where the least upper bound is taken over all
\(x\in e\cap g\cap\beta\);

\[ d(e,g,\beta_p)=\{\operatorname{mes}_p(g\cap\beta_p)-\operatorname{mes}_p(e\cap g\cap\beta_p)\}^{1/p}, \]

where \(\beta_p \subset E_n\) is a \(p\)-dimensional plane;

\[ m(e,g,\beta)=\min\{[\Gamma(e,g,\beta)],\ [d(e,g,\beta)]\}, \]

\[ k(g,\beta)=\Gamma(g,g,\beta)[d(g)]^{-1}. \]

Lemma 1. Let there be given a convex domain \(g \subset E_n\) such that \(k(g,E_n)\geq \gamma>0\), and a set \(e\subset E_n\) such that every component of the set \(e\cap g\) meets \(\partial(g\cap E_n)\).

Denote by \(\Omega(\Delta_1,\Delta_2,g)\) the set of hyperplanes \(\beta_{n-1}\) determined by the conditions: \(k(g,\beta_{n-1})\geq \Delta_1\) and \(m(e,g,\beta_{n-1})\geq \Delta_2 m(e,g,E_n)\).

There exist numbers \(\varepsilon>0\), \(\Delta_1>0\), and \(\Delta_2>0\), depending only on \(n\) and \(\gamma\), such that

\[ \mu_n^1[\Omega(\Delta_1,\Delta_2,g)]\geq \varepsilon m(e,g,E_n). \]

Lemma 2. Let the set \(e\subset E_n\) be closed and bounded, and let \(g_1,g_2,\ldots,g_p\) be convex, pairwise disjoint sets, with

\[ k(g_i,E_n)\geq \gamma>0\quad (i=1,2,\ldots,p). \]

Then for every \(k=0,1,2,\ldots,n-1\),

\[ \mathbf{V}_k(e)\geq \sum_{i=1}^{p} C_{i,k}[m(e,g_i,E_n)]^k, \]

where \(\{C_{i,k}\}\) are nonnegative numbers such that, for every \(i\),

\[ \sum_{k=0}^{n-1} C_{i,k}\geq C(n,\gamma)>0 \]

(\(C(n,\gamma)\) depends only on \(n\) and \(\gamma\)).

Proof. The lemma is proved by induction on \(n\).

Consider the case \(n=1\). Among all intervals \(\{g_i\}\) intersecting one and the same component of the set \(e\), at most two intervals can satisfy the condition \(m(e,g_i,E_1)>0\), and therefore

\[ \mathbf{V}_0(e)\geq \sum_{i=1}^{p} [m(e,g_i,E_1)]^0\cdot \frac{1}{2}. \]

(Here it is assumed that if \(m(e,g,E_1)=0\), then \([m(e,g_i,E_1)]^0=0\)); that is, for \(n=1\) the lemma is proved.

Now consider the general case. Recall that

\[ \mathbf{V}_k(e)=C(n,k)\int_{\Omega_n^1}\mathbf{V}_{k-1}(e\cap \beta_{n-1})\,d\mu_n^1, \]

where \(C(n,k)>0\) is a normalizing constant independent of \(e\). By the induction hypothesis, for every plane \(\beta_{n-1}\) we have

\[ \mathbf{V}_{k-1}(e\cap \beta_{n-1})\geq \sum_i C_{i,k-1}(\beta_{n-1})[m(e,g_i,\beta_{n-1})]^{k-1}, \]

\[ \sum_{q=1}^{n-2} C_{i,q}(\beta_{n-1})\geq C((n-1),\gamma)\quad (i=1,2,\ldots,p). \]

Here we assume that if \(g_i\cap \beta_{n-1}\) is empty, then \(m(e,g_i,\beta_{n-1})=0\).

Integrating the last inequality, we obtain

\[ \frac{\mathbf{V}_k(e)}{C(n,k)}\int_{\Omega_n^1}\mathbf{V}_{k-1}(e\cap \beta_{n-1})\,d\mu_n^1 \geq \sum_i \int_{\Omega_n^1} C_{i,k-1}(\beta_{n-1})[m(e,g_i,\beta_{n-1})]^{k-1}\,d\mu_n^1. \]

Put

\[ C_{i,k}=C(n,k)[m(e,g_i,E_n)]^{-k} \int_{\Omega_n^1} C_{i,k-1}(\beta_{n-1})[m(e,g_i,\beta_{n-1})]^{k-1}\,d\mu_n^1 \]

\[ (k=1,2,\ldots,n-1);\qquad C_{i,0}=1,\quad \text{if } g_i\cap e \text{ contains a component meeting} \]

\(\partial(g_i,E_n)\) components, and \(C_{i,0}=0\) otherwise; we have

\[ V_0(e)\geq \sum_i C_{i,0} =\sum_i C_{i,0}\,[m(e,g_i,E_n)]^0. \]

Thus, for every \(k=0,1,\ldots,n-1\),

\[ V_k(e)\geq \sum_i C_{i,k}\,[m(e,g_i,E_n)]^{k-1}, \]

and, by virtue of Lemma 1,

\[ \begin{aligned} \sum_{k=0}^{n-1} C_{i,k} &= C_{i,0}+\sum_{k=1}^{n-1} C(n,k) \int_{\Omega(\Delta_1,\Delta_2,g_i)} C_{i,k-1}(\beta_{n-1})[m(e,g_i,\beta_{n-1})]^{k-1}\times\\ &\quad \times [m(e,g_i,E_n)]^{-k}\,d\mu_n^1 \geq C_{i,0}+\sum_{k=1}^{n-1} C(n,k)[m(e_i g_i,E_n)]^{-k}\times\\ &\quad \times \int_{\Omega(\Delta_1,\Delta_g,g_i)} C_{i,k-1}(\beta_{n-1})[\Delta_2 m(e,g_i,E_n)]^{k-1}\,d\mu_n^1 \geq\\ &\geq C_{i,0}+\sum_{k=1}^{n-1} C(n,k)\Delta_2^{k-1}[m(e,g_i,E_n)]^{-1} \int_{\Omega(\Delta_1,\Delta_2,g_i)} C_{i,k}(\beta_{n-1})\,d\mu_n^1 \geq\\ &\geq C_{i,0}+\min_k C(n,k)\Delta_2^{k-1}[m(e,g_i,E_n)]^{-1} \int_{\Omega(\Delta_1,\Delta_2,g_i)}\sum_{k=1}^{n-k} C_{i,k}(\beta_{n-1})\,d\mu_n^1 \geq\\ &\geq C_{i,0}+C'(n,\Delta_1)[m(e,g_i,E_n)]^{-1}\varepsilon\cdot m(e,g_i,E_n)C((n-1),\Delta_1,\gamma)\geq C(n,\gamma). \end{aligned} \]

The lemma is proved.

Lemma 3. Let the closed set \(e\) be an \(\varepsilon\)-net for the measurable set \(f\subset E_n\). Then, for every \(\varepsilon\leq \varepsilon(f)\), for some \(k=k(\varepsilon)\) the inequality
\[ V_k(e)\geq C(n)[\operatorname{mes}_n f-\operatorname{mes}_n e]\varepsilon^{k-n}, \]
holds, where \(C(n)\) depends only on \(n\).

Proof. We shall assume that \(\Delta=\operatorname{mes}_n f-\operatorname{mes}_n e>0\). Fix \(p\) equal pairwise nonintersecting cubes \(g_1,g_2,\ldots,g_p\) with side \(4\varepsilon\) and such that

\[ \sum_i \operatorname{mes}_n(f\cap g_i)\geq (1-\tfrac18\Delta)\operatorname{mes}_n f. \]

Since
\[ \sum_i [m(e,g_i,E_n)]^n\geq C_1(n)\Delta, \]
then by virtue of Lemma 2 one can indicate indices \(i_1,i_2,\ldots,i_q\) and a number \(k\) such that

\[ V_k(e)\geq \frac{C(n,\gamma)}{n}\sum_{s=1}^{q}[m(e,g_{i_s},E_n)]^k, \]

\[ \sum_{s=1}^{q} m(e,g_{i_s},E_n)^n\geq C_2(n)\Delta, \]

and consequently,
\[ V_k(e)\geq C(n)\cdot \Delta\cdot \varepsilon^{k-n}. \]

The lemma is proved.

Notation.
Let \(\varphi_n^k\) be the bundle of \(k\)-dimensional planes \(\tau_k\) passing through one and the same point; let \(\overline m_n^k\) be the measure in the space \(\varphi_n^k\), invariant with respect to the group of transformations generated by rotations of space; \(\beta_{s+n-k}(\beta_s,\tau_k)\) be the plane of dimension \(s+n-k\) containing the \(s\)-dimensional plane \(\beta_s\subseteq\tau_k\) and the \((n-k)\)-dimensional plane \(\beta_{n-k}\subset E_n\), which is ...

orthogonal complement to \(\tau_k\)

\[ V_{s,\tau_k}(e)=\int_{\beta_s\subset\tau_k} V_0\bigl(e\cap \beta_{s+n-k}(\beta_s,\tau_k)\bigr)\,d\mu_k^{\,k-s}. \]

Lemma 4. Let the set \(f\subset E_n\) be closed and such that, for every plane \(\beta_s\subset\tau_k\) (\(\tau_k\) fixed) and for every pair of components \(f_1\) and \(f_2\) of the set \(f\cap \beta_{s+n-k}(\beta_s,\tau_k)\),

\[ \min_{\substack{a\in f_1\\ b\in f_2}}\rho(a,b)\ge \Delta>0 \]

(\(\Delta\) does not depend on \(\beta_s,f_1,f_2\)), and let the set \(e\) be an \(\varepsilon\)-net for \(f\) and lie in the \((\tfrac14\Delta)\)-neighborhood of the set \(f\). Then, if \(\varepsilon<\varepsilon(f)\) (\(\varepsilon(f)\) does not depend on \(e\)), then

\[ \sum_{s=0}^{k-1} V_{s,\tau_s}(e)\ge C(k)\,[V_{k,\tau_k}(f)-V_{k,\tau_k}(e)]\,\varepsilon^{-1} \]

(\(C(k)\)—see Lemma 3).

Lemma 5. If a closed set is equal to \(\lim_{i\to\infty} e_i\), then for every plane \(\tau_k\)

\[ \lim_{i\to\infty}\sum_{s=0}^{k-1} V_{s,\tau_k}(e_i)=+\infty, \]

provided only that

\[ \inf_{i\to\infty} V_{k,\tau_k}(e_i)<V_{k,\tau_k}(e). \]

The lemma follows easily from the preceding lemma, and from it the proof of Theorem 1 is obtained, since

\[ V_s(f)=C(n,k,s)\int_{\varphi_n^k} V_{s,\tau_k}(e)\,dm_n^k, \]

where \(C(n,k,s)\) is a normalizing factor independent of the set \(f\).

Received
15 VI 1965

REFERENCES

1. A. G. Vitushkin, On multidimensional variations, Moscow, 1955.