Classifying invariant $\sigma $-ideals with analytic base on good Cantor measure spaces

Abstract

Abstract. Let X be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel σ-additive measure µ which is good in the sense that for any clopen subsets U, V ⊂ X with µ(U ) < µ(V ) there is a clopen set W ⊂ V with µ(W ) = µ(U ). We study σ-ideals with Borel base on X which are invariant under the action of the group Hµ(X) of measure-preserving homeomorphisms of (X, µ), and show that any such σ-ideal I is equal to one of seven σ-ideals:Here [X] ≤κ is the ideal consisting of subsets of cardiality ≤ κ in X, M is the ideal of meager subsets of X, N = {A ⊂ X : µ(A) = 0} is the ideal of null subsets of (X, µ), and E is the σ-ideal generated by closed null subsets of (X, µ).

Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses

“…Consider the projection A of A onto X <n . 6 5 µ(A) and hence 0 < 5 6 λ (A ) < µ(A). Consider the set…”

“…The equivalence of ( 1)-( 4) was proved in Theorem 19.3. The implications (3)⇒ (5,6) are trivial. To finish the proof, it suffices to show that the negation of (1) implies the negations of ( 5) and (6).…”

“…The equivalence of the conditions ( 1)-( 3) follows from Theorem 18.1, (1)⇔(4) follows from Theorem 9.7, and the implications (3)⇒(5, 6) are trivial. To finish the proof it suffices to show that the negation of (1) implies the negations of ( 5) and (6). So, assume that the subgroup A − A is open in X.…”

“…On the other hand, for any strictly positive continuous measure µ on a Polish space X, the σ-ideal σN µ is a proper subideal of M X ∩ N µ . In [6], [8] the σ-ideal σN on the Cantor cube is denoted by E. Now we recall some known Steinhaus-type properties of the ideals M and N on topological groups. Theorem 3.3 (Steinhaus [53], Weil [55, §11]).…”

“…We lose no generality assuming that the sets A, B are contained in n∈ω Ω n . By the regularity of the measure µ, there exists a neighborhood U ⊆ X of θ such that µ(A + U ) < 6 5 µ(A) and µ(B + U ) < 6 5 µ(B). Replacing U by a smaller neighborhood, we can assume that U is of the basic form U = V + X ≥n for some n ∈ ω and some open set V ⊆ X <n .…”