Chacon transformation is ergodic

Fix $0<\delta <1$. At every sufficiently late stage r, F2 supplies a level J with $\mu (E\cap J)>(1-\delta )w_r$. Repeatedly composing F5's consecutive partial translations shows that the finite partial map sends level j to level k after $k-j$ iterates whenever $j\le k<h_r$. F1 makes the limiting T agree with these arrows on its invariant conull set. Strict invariance implies equality of the measures of E in these levels, since the iterates preserve measure and membership in E. Removing the fixed null complement does not affect these equalities. Hence every level of this tower has E-measure greater than $(1-\delta )w_r$.

Summing over the disjoint levels gives $\mu (E)\ge \mu (E\cap C_r)>(1-\delta )\mu (C_r)$. Letting $r\to \mathrm{\infty }$ gives $\mu (E)\ge 1-\delta$ because $\mu (C_r)\to 1$. Since every $0<\delta <1$ is allowed and $\mu (E)\le 1$, $\mu (E)=1$. Sets of zero measure already satisfy the alternative. This is ergodicity by F3. AC is inherited from the measure construction and generating-level approximation, with only one finite-stage level needed at a time.