Part of the book series: Universitext ((UTX))
Suppose a probability measure Q is given on a product space \(\varOmega = \mathop {\prod }_{t\in \varLambda } S_{t}\) with the product \(\sigma \)-field \(\mathcal {F}= \otimes _{t \in \varLambda } \mathcal {S}_t\).
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This proof is due to Edward Nelson (1959), Regular Probability Measures on Function Spaces, Ann. of Math. 69, 630–643.
See Appendix B for a proof of Tychonoff’s theorem for the case of countable \(\varLambda \). For uncountable \(\varLambda \), see Folland (1984).
For general locally compact Hausdorff spaces see Folland (1984), or Royden (1988).
For a proof of Tulcea’s theorem see Ethier and Kurtz (1986), or Neveu (1965).
See e.g., Bhattacharya, R. N. and Waymire E.C. (2016), Stationary Processes and Discrete Parameter Markov Processes, Chapter I, Sec 2, Springer (to appear).
This construction originated in Ciesielski, Z. (1961): Hölder condition for realization of Gaussian processes, Trans. Amer. Math. Soc. 99 403–413, based on a general approach of Lévy, P. (1948), p. 209.
© 2016 Springer International Publishing AG
Bhattacharya, R., Waymire, E.C. (2016). Kolmogorov’s Extension Theorem and Brownian Motion. In: A Basic Course in Probability Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-47974-3_9
DOI: https://doi.org/10.1007/978-3-319-47974-3_9
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-47972-9
Online ISBN: 978-3-319-47974-3
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