An article consists of two parts.

In the first part the sufficient and necessary conditions for an integral representation of hyperbolically convex (h.c.) functions $k(x)$ $\left(x\in \mathbb{R}^{\infty}= \mathbb{R}^1\times\mathbb{R}^1\times \dots\right)$ are proved. For this purpose in $\mathbb{R}^{\infty}$ we introduce measures $\omega_1(x)$, $\omega_{\frac{1}{2}}(x)$. The positive definiteness of a function will be understood on the integral sense with respect to the measure $\omega_1(x)$. Then we proved that the measure $\rho(\lambda)$ in the integral representation is concentrated on $l_2^+=\bigg\{\lambda \in \mathbb{R}_+^{\infty}= \mathbb{R}_+^1\times\mathbb{R}_+^1\times \dots\Big|\sum\limits_{n=1}^{\infty}\lambda_n^2<\infty\bigg\}$. The equality for $k(x)$ $\left(x\in\mathbb{R}^{\infty} \right)$ is regarded as an equality for almost all $x\in\mathbb{R}^{\infty}$ with respect to measure $\omega_{\frac{1}{2}}(x)$.

In the second part we proved the sufficient and necessary conditions for integral representation of h.c. functions $k(x)$ $\big(x\in \mathbb{R}_0^{\infty}$ $\mathrm{~is~a~nuclear~space}\big)$. The positive definiteness of a function $k(x)$ will be understood on the pointwise sense. For this purpose we shall construct a rigging (chain) $\mathbb{R}_0^{\infty}\subset l_2 \subset \mathbb{R}^{\infty}$. Then, given that the projection and inductive topologies are coinciding, we shall obtaine the integral representation for $k(x)$ $\left(x\in \mathbb{R}_0^{\infty}\right)$

[1] Berezansky Yu. M. Expansions in eigenfunctions of self-adjoint operators. Translations of Mathematical
Monographs Vol. 17, Providence, R.I.: Am. Math. Soc., 1968, 809 p.
[2] Berezansky Yu. M., Gali I. M. Positive definite functions of infinite many variables in a layer.
Ukr. Math. J. 1972, 24 (4), 351–372. doi:10.1007/BF01314686.
[3] Berezansky Yu. M. Self-adjoint operators in space of functions of infinitely many varibles. Kyiv, Naukova
dumka, 1978.
[4] Berezansky Yu. M., Kalyuzhny A. A. Representation of hypercomplex systems with locally compact
basis. Ukr. Math. J. 1984, 36 (4), 417–421. doi:10.1007/BF01066549.
[5] Berezansky Yu. M., Kondratiev Yu. G. Spectral methods in infinite-dimensional analysis. Kyiv,
Naukova dumka, 1988.
[6] Bogolyubov N. N., Logunov A. A., Oksak A. I., Togorov I. T. General Principles of Quantum Field
Theory. Moskov, Nauka, 1987.
[7] Lopotko O. V., Rudinski I. I. Integral representation of evenly positive-definite bounded functions of
infinite number of variables. Ukr. Math. J. 1982, 34 (3), 310–312. doi:10.1007/BF01682127.
[8] Lopotko O. V. Even positive definite bounded functions of infinitely many variables. Dokl. AN of
Ukraine, Ser. A. 1991, 8, 11–13.
[9] Lopotko O. V. The integral representation for odd positive definite functions of infinitely many variables.
Dokl. AN of Ukraine, 2006,7, 11–13.
[10] Lopotko O. V. The integral representation of positively definite kernels of finite and infinite many
variables. Ph.D. Inst. of Math. Kiev, 1992.
[11] Rudinsky I. I. The integral representation for evenly positive-definite functions on nuclear space.
Ukr. Math. J. 1984, 36 (4), 429–431. doi:10.1007/BF01066570.
[12] Halmos P. R. Measure Theory. Moskov, Publishing House of Foreign Literature, 1953.
[13] Schaefer H. Topological Vector Spaces. Moskov, Peace, 1971.