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

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