Abstract
The main purpose of this note is to give a simpler and more general definition of “weak” or “weak-star” convergence of certain measures on non-separable metric spaces, and to prove its equivalence with the convergence introduced in [1] for the cases considered there.
Received July 1, 1966.
This research was supported in part by a National Science Foundation grant, and was presented to the International Congress of Mathematicians in Moscow, August 1966.
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References
R. M. Dudley, Weak convergence of probabilities on nonseparable metric spaces and empirical measures on Euclidean spaces, Illinois J. Math., vol. 10 (1966), pp. 109–126.
E. Marczewski and P. Sikorski, Measures in nonseparable metric spaces, Colloq. Math., vol. 1 (1948), pp. 133–139.
H. Stone, Cardinals of closed sets, Mathematika, vol. 6 (1959), pp. 99–107.
S. Ulam and J. C. Oxtoby, On the existence of a measure invariant under a transformation, Ann. of Math., vol. 40 (1939), pp. 560–566.
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Dudley, R.M. (2010). Measures on Non-Separable Metric Spaces. In: Giné, E., Koltchinskii, V., Norvaisa, R. (eds) Selected Works of R.M. Dudley. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5821-1_3
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