## Characterization of conditional expectation operators for Banach-valued functions

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- by D. Landers and L. Rogge PDF
- Proc. Amer. Math. Soc.
**81**(1981), 107-110 Request permission

## Abstract:

Let $B$ be a Banach space. It is well known that for every probability space $(\Omega , \mathcal {A}, P)$ and every sub-$\sigma$-field $\mathcal {A}_0 \subset \mathcal {A}$ there exists a conditional expectation operator $P^{\mathcal {A}_0}X$ for $B$-valued $P$-integrable functions. This operator maps the space ${L_p}(\Omega , \mathcal {A}, P, B)$ into itself for each $p \geqslant 1$. The operator is linear, idempotent, constant preserving and contractive in ${L_p}$. For $B = {\mathbf {R}}$ and $p \ne 2$ these conditions characterize a conditional expectation operator. It turns out that in general these properties characterize conditional expectation operators for Banach-valued functions only for $p = 1$ and strictly convex Banach spaces.## References

- R. R. Bahadur,
*Measurable subspaces and subalgebras*, Proc. Amer. Math. Soc.**6**(1955), 565β570. MR**72446**, DOI 10.1090/S0002-9939-1955-0072446-8 - J. Diestel and J. J. Uhl Jr.,
*Vector measures*, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR**0453964** - R. G. Douglas,
*Contractive projections on an ${\mathfrak {L}}_{1}$ space*, Pacific J. Math.**15**(1965), 443β462. MR**187087** - Harro Heuser,
*Funktionalanalysis*, Mathematische LeitfΓ€den, B. G. Teubner, Stuttgart, 1975 (German). MR**0482021** - Shu-Teh Chen Moy,
*Characterizations of conditional expectation as a transformation on function spaces*, Pacific J. Math.**4**(1954), 47β63. MR**60750** - J. Neveu,
*Discrete-parameter martingales*, Revised edition, North-Holland Mathematical Library, Vol. 10, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed. MR**0402915** - J. Pfanzagl,
*Characterizations of conditional expectations*, Ann. Math. Statist.**38**(1967), 415β421. MR**211430**, DOI 10.1214/aoms/1177698957 - M. M. Rao,
*Smoothness of Orlicz spaces. I, II*, Nederl. Akad. Wetensch. Proc. Ser. A 68 = Indag. Math.**27**(1965), 671β680; 681β690. MR**0190704** - M. M. Rao,
*Inference in stochastic processes. III. Nonlinear prediction, filtering, and samlping theorems*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**8**(1967), 49β72. MR**216713**, DOI 10.1007/BF00533944

## Additional Information

- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**81**(1981), 107-110 - MSC: Primary 60B11; Secondary 60A05
- DOI: https://doi.org/10.1090/S0002-9939-1981-0589148-7
- MathSciNet review: 589148