## Real Analysis Exchange

### On Marczewski-Burstin Like Characterizations of Certain σ-Algebras and σ-Ideals

Hussain Elalaoui-Talibi

#### Abstract

Consider a $\sigma$-ideal, $\sigma$-algebra pair $\cal{I} \subseteq \cal{A}$ on a Polish space $X$ which has no isolated points, such that $\cal{A}$ contains all the Borel subsets of $X$ while $\cal{I}$ contains all the countable subsets of $X$, but none of the perfect subsets of $X$. We show that if $(\cal{I}, \cal{A})$ admits a simultaneous MB-like characterization consisting of Borel sets, then $(\cal{I},\cal{A})$ is $((s_{0}), (s))$, the $\sigma$-ideal, $\sigma$-algebra pair of Marczewski null, Marczewski measurable sets. We deduce some results about uniformly completely Ramsey sets.

#### Article information

Source
Real Anal. Exchange, Volume 26, Number 1 (2000), 413-416.

Dates
First available in Project Euclid: 2 January 2009

https://projecteuclid.org/euclid.rae/1230939169

Mathematical Reviews number (MathSciNet)
MR1825519

Zentralblatt MATH identifier
1009.28002

#### Citation

Elalaoui-Talibi, Hussain. On Marczewski-Burstin Like Characterizations of Certain σ-Algebras and σ-Ideals. Real Anal. Exchange 26 (2000), no. 1, 413--416. https://projecteuclid.org/euclid.rae/1230939169

#### References

• Brown, J. B., Elalaoui-Talibi, H. (1999) Marczewski-Burstin-like characterizations of $\sigma$-algebras, ideals, and measurable functions. Coll. Math. 82, (1999), 277-286.
• Darji, U. B. (1993) Uniformly completely Ramsey sets. Coll. Math. 64, (1993), 163-171.
• Walsh, J. (1989) Marczewski sets, measure and the Baire property, II. Proc. Amer. Math. Soc. 106 (1989), 1027-1030.