Geodesic
On a~sufficient condition for integrability of the banach indicatrix of a~smooth mapping
Izvestiya. Mathematics , Tome 23 (1984) no. 2, pp. 391-408.

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Sufficient conditions are given for the convergence of integrals of the form $\int_{\mathbf R^k}(N(f,y))^p\,dy$, where $f\colon\mathbf R^n\to\mathbf R^k$ is a map with compact support with $k\leqslant n$ and $N(f,y)$ is the number of connected components of the level set $\{x:f(x)=y\}$. Bibliography: 5 titles. 
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S. A. Gulevich. On a~sufficient condition for integrability of the banach indicatrix of a~smooth mapping. Izvestiya. Mathematics , Tome 23 (1984) no. 2, pp. 391-408. http://geodesic.mathdoc.fr/item/IM2_1984_23_2_a7/

[1] Ivanov L. D., Variatsii mnozhestv i funktsii, Nauka, M., 1975 | MR

[2] Vitushkin A. G., O mnogomernykh variatsiyakh, GTTI, M., 1955

[3] Khirsh M., Differentsialnaya topologiya, Mir, M., 1979 | MR | Zbl

[4] Federer H., Geometric measure theory, Springer, Berlin, 1969 | MR

[5] Ivanov L. D., Elementy polilineinoi algebry i ee prilozheniya k integrirovaniyu, KKhU, Kalinin, 1977