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February, 1979 Conditional Expectation and Ordering
Ep de Jonge
Ann. Probab. 7(1): 179-183 (February, 1979). DOI: 10.1214/aop/1176995162


Let $(\Omega, \mathscr{A}, \mu)$ be a probability space and let $L$ be an ideal in $M(\Omega, \mathscr{A}, \mu)$ containing $\chi_\Omega$. A one-one correspondence between the class of "order closed" linear subspaces of $L$ and the sub $\sigma$-algebras of $\mathscr{A}$ is proved. Furthermore, if $T : L \rightarrow M(\Omega, \mathscr{A}, \mu)$ is a strictly positive order continuous projectionlike linear map then $T$ is shown to be a conditional expectation $E_\nu(\cdot \mid\mathscr{A}_0)$. It follows that if $T: L \rightarrow M(\Omega, \mathscr{A}, \mu)$ is a positive expectation invariant projectionlike linear map, then even $T = E_\mu(\cdot \mid \mathscr{A}_0)$.


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Ep de Jonge. "Conditional Expectation and Ordering." Ann. Probab. 7 (1) 179 - 183, February, 1979.


Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60004
MathSciNet: MR515827
Digital Object Identifier: 10.1214/aop/1176995162

Primary: 60A05
Secondary: 46E30 , 47B55

Keywords: conditional expectation , measurable subspace

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 1 • February, 1979
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