A remark on a paper of B. R. Gelbaum

Summary

In this paper the following generalization of a theorem by B.R. Gelbaum is proved:

Let (K, d) be a compact, connected metric space. Let B denote the Borel sets of (K, d), and P be a probability measure on B with P(G)≠O for any nonempty open G, f, gεC(K,d) independent random variables on (K,B,P) and let g satisfy the following assumption:

There is an y o∃ℝ such that g −1(y0) is a finite nonempty set. Then f is a constant function.

Examples show that the assumptions of this theorem are essential.

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Reference

  1. 1.

    Gelbaum, B.R.: Independence of events and of random variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete 36, 333–343 (1976)

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Bosznay, A.P. A remark on a paper of B. R. Gelbaum. Z. Wahrscheinlichkeitstheorie verw Gebiete 43, 353–355 (1978). https://doi.org/10.1007/BF00534768

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Keywords

  • Stochastic Process
  • Probability Measure
  • Probability Theory
  • Mathematical Biology
  • Constant Function