On some relations between properties of invariant $\sigma$-ideals in Polish spaces

In this paper we shall consider a couple of properties of $\sigma$-ideals and study relations between them. Namely we will prove that $\mathfrak{c}$-cc $\sigma$-ideals are tall and that the Weaker Smital Property implies that every Borel $\mathcal{I}$-positive set contains a witness for non($\mathcal{I}$) as well, as satisfying ccc and Fubini Property. We give also a characterization of nonmeasurability of $\mathcal{I}$-Luzin sets and prove that the ideal $[\mathbb{R}]^{\leq\omega}$ does not posses the Fubini Property using some interesting lemma about perfect sets.