# Robert Rałowski: Nonmeasurable unions with respect to analytic families

07/10/20 19:24

Tuesday, October 13, 2020 17:15

1) \((X,\rho)\) is separable metric space,

2) identity \(id:(X,\rho) \to X\) is continuous,

2) every \(\rho\)-Cauchy sequence is converged in \(X.\)

Family \(\mathcal A \subseteq P(X)\) is analytic if

1) \(X\in \mathcal A\)

2) \(\mathcal A\) is closed on intersections

3) each \(A\in{\mathcal A}\) has analytic metric \(\rho\) and for any \(\epsilon>0\) there is a countable cover \(\mathcal U \subseteq \mathcal A\) of \(\mathcal A\) with \(\epsilon\) \(\rho\)-diameter.

We present a theorem that generalizes the well known result obtained by Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski about nonmeasurable unions.

Theorem. Let \({\mathcal A}\) be an analytic family of Hausdorff space \(X\), any \(I\) \(\sigma\)-ideal of \(X.\) If \( J\subseteq I\) is point-finite family such that \(\bigcup J \notin I\) then there is a subfamily \(J' \subseteq J\) and \(A\in {\mathcal A}\) such that

1) \(A\cap \bigcap J' \notin I\)

2) for every \(A' \in {\mathcal A}\) if \(A' \subseteq A\cap \bigcup J'\) then \(A' \in I.\)

We show that the above Theorem implies the Theorem on nomeasurabie unions with respect to tree ideals like Marczeski ideal \(s_0\) for example.

Moreover, the above Theorem implies theorem on nonmeasurable unions with respect to \(\sigma\)-ideals which has Marczewski-Burstin representation.

The last mentioned result gives a theorem about nonmeasurable unions with respect to the ideal of Ramsey-null set in Ramsey space with Ellentuck topology.

The talk is based on a joint work with Taras Banakh and Szymon Żeberski.

*Location:***Zoom.us**: if you want to participate please contact organizers*Robert Rałowski*

Speaker:Speaker:

*Title*: Nonmeasurable unions with respect to analytic families*Abstract*: We say that metric \(\rho\) is analytic on Hausdorff topological space if1) \((X,\rho)\) is separable metric space,

2) identity \(id:(X,\rho) \to X\) is continuous,

2) every \(\rho\)-Cauchy sequence is converged in \(X.\)

Family \(\mathcal A \subseteq P(X)\) is analytic if

1) \(X\in \mathcal A\)

2) \(\mathcal A\) is closed on intersections

3) each \(A\in{\mathcal A}\) has analytic metric \(\rho\) and for any \(\epsilon>0\) there is a countable cover \(\mathcal U \subseteq \mathcal A\) of \(\mathcal A\) with \(\epsilon\) \(\rho\)-diameter.

We present a theorem that generalizes the well known result obtained by Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski about nonmeasurable unions.

Theorem. Let \({\mathcal A}\) be an analytic family of Hausdorff space \(X\), any \(I\) \(\sigma\)-ideal of \(X.\) If \( J\subseteq I\) is point-finite family such that \(\bigcup J \notin I\) then there is a subfamily \(J' \subseteq J\) and \(A\in {\mathcal A}\) such that

1) \(A\cap \bigcap J' \notin I\)

2) for every \(A' \in {\mathcal A}\) if \(A' \subseteq A\cap \bigcup J'\) then \(A' \in I.\)

We show that the above Theorem implies the Theorem on nomeasurabie unions with respect to tree ideals like Marczeski ideal \(s_0\) for example.

Moreover, the above Theorem implies theorem on nonmeasurable unions with respect to \(\sigma\)-ideals which has Marczewski-Burstin representation.

The last mentioned result gives a theorem about nonmeasurable unions with respect to the ideal of Ramsey-null set in Ramsey space with Ellentuck topology.

The talk is based on a joint work with Taras Banakh and Szymon Żeberski.