Arithmetic of (independent) sigma-fields on probability spaces
Volume 6, Issue 3 (2019), pp. 269–284
Pub. online: 20 June 2019
Type: Research Article
Open Access
Open Access
Received
19 January 2019
19 January 2019
Revised
17 May 2019
17 May 2019
Accepted
17 May 2019
17 May 2019
Published
20 June 2019
20 June 2019
Abstract
This note gathers what is known about, and provides some new results concerning the operations of intersection, of “generated σ-field”, and of “complementation” for (independent) complete σ-fields on probability spaces.
References
Burkholder, D.L., Chow, Y.S.: Iterates of conditional expectation operators. Proc. Am. Math. Soc. 12(3), 490–495 (1961). MR0142144. https://doi.org/10.2307/2034224
Chaumont, L., Yor, M.: Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, Via Conditioning. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press (2003). MR2016344. https://doi.org/10.1017/CBO9780511610813
Émery, M., Schachermayer, W.: A remark on Tsirelson’s stochastic differential equation. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds.) Séminaire de Probabilités XXXIII, pp. 291–303. Springer, Berlin, Heidelberg (1999). MR1768019. https://doi.org/10.1007/BFb0096508
Émery, M., Schachermayer, W.: On Vershik’s standardness criterion and Tsirelson’s notion of cosiness. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds.) Séminaire de Probabilités XXXV, pp. 265–305. Springer, Berlin, Heidelberg (2001). MR1837293. https://doi.org/10.1007/978-3-540-44671-2_20
Feldman, J.: Decomposable processes and continuous products of probability spaces. J. Funct. Anal. 8(1), 1–51 (1971). MR0290436. https://doi.org/10.1016/0022-1236(71)90017-6
Kallenberg, O.: Foundations of Modern Probability. Probability and Its Applications. Springer, New York Berlin Heidelberg (1997). MR1464694
Lindvall, T., Rogers, L.C.G.: Coupling of multidimensional diffusions by reflection. Ann. Probab. 14(3), 860–872 (1986). MR0841588
Meyer, P.A.: Probability and Potentials. Blaisdell book in pure and applied mathematics. Blaisdell Publishing Company (1966). MR0205288
Neveu, J., Speed, T.P.: Discrete-parameter Martingales. North-Holland mathematical library. North-Holland (1975). MR0402915
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften. Springer (2004). MR1083357. https://doi.org/10.1007/978-3-662-21726-9
Schramm, O., Smirnov, S., Garban, C.: On the scaling limits of planar percolation. Ann. Probab. 39(5), 1768–1814 (2011). MR2884873. https://doi.org/10.1214/11-AOP659
Tsirelson, B.: Noise as a Boolean algebra of σ-fields. Ann. Probab. 42(1), 311–353 (2014). MR3161487. https://doi.org/10.1214/13-AOP861
Weizsäcker, H.V.: Exchanging the order of taking suprema and countable intersections of σ-algebras. Ann. I.H.P. Probab. Stat. 19(1), 91–100 (1983). MR0699981
Williams, D.: Probability with Martingales. Cambridge mathematical textbooks. Cambridge University Press (1991). MR1155402. https://doi.org/10.1017/CBO9780511813658
Yor, M.: Tsirel’son’s equation in discrete time. Probab. Theory Relat. Fields 91(2), 135–152 (1992). MR1147613. https://doi.org/10.1007/BF01291422
