Let {A_n} be a sequence of sets, limsup A_n=, liminf A_n and lim A_n are defined in probability theory.
I would like to know who introduced this notions and find some papers on this context!
Thanks so much!
Dear professor Cong Dan Pham
The definition in you question coincide with the definition of Kuratowski for sets. See K.Kuratowski. Topology v.1 &1.V. He not use any metric.The definitions with metric Painleve, C.R. Paris v.148 (1909) and Zoretti J.Math. v.1 (1905) see K.Kuratowski. Topology v.1 &29. I.
Sincerelly Yuri
"Set-valued Analysis" of Jean-Pierre Aubin and Helene Frankowska seems to be a good ref: http://www.springer.com/us/book/9780817648473
This is Kuratowski konvergence
https://en.wikipedia.org/wiki/Kuratowski_convergence
Thank you so much Sir!
I think it seems that the definition of Kuratowski is different because he need a metric in the definition. That is not the definition in my question?
Dear professor Cong Dan Pham
The definition in you question coincide with the definition of Kuratowski for sets. See K.Kuratowski. Topology v.1 &1.V. He not use any metric.The definitions with metric Painleve, C.R. Paris v.148 (1909) and Zoretti J.Math. v.1 (1905) see K.Kuratowski. Topology v.1 &29. I.
Sincerelly Yuri
Hello,
I think there is not a unique answer to your question. However :
- Fréchet in his thesis in 1906 introduced the bases for topology over general sets and went on with several works on abstract sets in the following years
- the liminf definition may be due to Borel in his 1909 paper introducing the strong law of large numbers by means of (today's name of course) Borel-Cantelli lemma expressing that only a finite number of events in a sequence take place.
Sincerely,
Laurent
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