Abstract
We develop a general method for the construction of a probability structure on the space F of random sets in ℝ. For this purpose, by using the introduced notion of c-system, we prove a theorem on the unique extension of a finite measure from a c-system to the minimal c-algebra. The obtained structure of measurability enables one to determine probability distributions of the c-algebra of random events sufficient, e.g., for the so-called fractal dimensionality of random realizations to be considered as a measurable functional on F.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1467–1483, November, 2004.
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Virchenko, Y.P., Shpilinskaya, O.L. Probability Space of Stochastic Fractals. Ukr Math J 56, 1748–1765 (2004). https://doi.org/10.1007/s11253-005-0149-z
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DOI: https://doi.org/10.1007/s11253-005-0149-z