Suppose a probability measureQ is given on a product space\(\varOmega = \mathop {\prod }_{t\in \varLambda } S_{t}\) with the product\(\sigma \)-field\(\mathcal {F}= \otimes _{t \in \varLambda } \mathcal {S}_t\).

1. 1.

   This proof is due to Edward Nelson (1959),Regular Probability Measures on Function Spaces, Ann. of Math.69, 630–643.

2. 2.

   See Appendix B for a proof of Tychonoff’s theorem for the case of countable\(\varLambda \). For uncountable\(\varLambda \), see Folland (1984).

3. 3.

   For general locally compact Hausdorff spaces see Folland (1984), or Royden (1988).

4. 4.

   For a proof of Tulcea’s theorem see Ethier and Kurtz (1986), or Neveu (1965).

5. 5.

   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).

6. 6.

   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.

Authors and Affiliations

1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

   Rabi Bhattacharya

2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

   Edward C. Waymire

Corresponding author

Correspondence toRabi Bhattacharya.

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