# On the Foundations of Probability Theory

Chapter

## Abstract

My paper will deal with some aspects of the foundations of objective probability theory, more specifically, of the frequency theory of objective probability (see p. 207 for a closer description of our aim). The word ‘objective’ is used in contrast to ‘subjective’ or ‘personal’, and ‘theory’ indicates that we do not plan an analysis of the daily-life use of the term ‘probability’.

## Keywords

Objective Probability 2Zoological Museum Nomen Nudum Queensland Museum Common Possession
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## References

- 1.L. Savage,
*The Foundations of Statistics*, New York 1954.Google Scholar - 2.In the light of the above discussion the reader may consider lines 5–14 on p. 4 of L. Savage’s above-quoted book. It should be clear that these critical remarks of Savage rest on misunderstandings that we have tried to clarify here.Google Scholar
- 3.The hyphen in this sentence is by H.G.Google Scholar
- 4.An excellent book on probability states: “The theory of probability is concerned with the measure properties of various spaces and with the mutual relations of measurable functions defined on these spaces.” Now we have it!Google Scholar
- 5.The often-used term ‘sample space’ belongs to theoretical statistics, where we have to distinguish between sample space and label space; there, sample space is the space of the observations and not of the probabilities.Google Scholar
- 6.A system
*T*of sets is called a field if together with two sets*A*and*B*of T the sum*A*+*B*, the intersection*AB*, and the difference*A*—*B*if it exists belongs to T. It is a*σ*-field if together with any enumerable system {*A*n} of sets of*T*also ∑*A*_{n}belongs to*T*.Google Scholar - 7.See Chapter One, Section 7 of R. von Mises,
*Mathematical Theory of Probability and Statistics*(edited and complemented by Hilda Geiringer), New York 1964. This book will be quoted as [1]Google Scholar - 8.I quote from a paper of Einstein: “Knowledge does not spring from experience alone but from the comparison of the inventions of the intellect with observed facts.”Google Scholar
- 9.Die Widerspruchsfreiheit des Kollektivbegriffs’,
*Ergebnisse eines mathem. Kolloquiums*8 (1937) 38–72.Google Scholar - 10.The same holds if there are more than two labels.Google Scholar
- 11.J. L. Doob,
*Annals of Mathematics*, 37 (1936) 363–367.CrossRefGoogle Scholar - 12.It is enough for our purpose to consider this Q; the results hold for any finite or countable set of labels.Google Scholar
- 13.One needs an assumption on si to the effect that for ‘almost all’ sequences co of Q the transformed co’ is infinite. Let us call this condition (/).Google Scholar
- 14.See A. Wald, loc. cit.9 p. 57, Satz 1.Google Scholar
- 15.
*Bull. Amer. Math. Soc*. 46 (1940) p. 130ff.; see his footnote 20, page 135.Google Scholar - 16.Von Mises was satisfied with Wald’s idea (the use of (L) as the set of selectors) and probably did not know the work of Church and of Turing. My attention was called to Church’s paper only recently through a quotations in a recent paper of Kolmogorov. I then obtained better understanding of these ideas through letters of Church and through conversations with the logician E. Storm.Google Scholar
- 17.This trivial misunderstanding occurs in a review of [1],
*Ann. Math. Statistics*, 37 (1966), where the fact that the regular sequence 010101… contains countably many subsequences in which the frequency limits of the labels exist and equal 1/2 is interpreted as implying that this sequence is a collective, with the purpose of demonstrating absurdity of the concept of collective.Google Scholar - 18.See p. 19 of [1]. See also, Hilda Geiringer, ‘Statistical Investigations of Transcendental Numbers’, in Studies presented to Richard von Mises, New York 1954, p. 31 Off.Google Scholar
- 19.Math. Assoc. of America, New York, 1959.Google Scholar
- 20.In the frequency definition of Pn we consider N groups, each of n throws; each group leads to a number
*x*of heads; we speak of ‘success’ if \x/n—p\ < ɛ, otherwise of ‘failure’. If there are among the*N*groups*N1*‘successes’ then P_{n}= lim_{N}→∞*N*1/*N*.Google Scholar - 21.The so-called
*strong law of large numbers*is a theorem in the general label space to be considered in Section III.Google Scholar - 22.The dependence on the valuation enters also into the definition of Lebesgue measurability.Google Scholar
- 23.It would lead to far to describe here the relation of Tornier’s work to our approach and to the ‘different structure’ mentioned in the text. A sketch of his important work is presented in [1], Appendix Three, pp. 98–112.Google Scholar

## Copyright information

© D. Reidel Publishing Company / Dordrecht-Holland 1967