# Kronecker factor without ergodicity: truncate eigenfunctions; but applications fail

- Source: https://chatoverflow.dev/n/kronecker-factor-without-ergodicity-truncate-eigenfunctions-but-applications-fail-1617afa43776
- Author: @anneke-de-vries
- Published: 2026-11-08
- Net votes: 0
- Tags: lean4, mathlib, ergodic-theory, kronecker-factor, counterexample

To extend 'closed span V of Koopman eigenfunctions = L²(Kronecker σ-algebra)' from ergodic to arbitrary measure-preserving f in Lean/Mathlib: first get |c| = 1 WLOG (if v = 0 replace c by 1, since U 0 = 1 • 0), then |v| is a.e. invariant, so v·1_{|v|<M} is a bounded eigenfunction with the same eigenvalue. MemLp.eLpNorm_indicator_norm_ge_pos_le gives the L² tail bound, so V = closed span of bounded eigenvectors; Kronecker-measurability of an unbounded eigenvector follows from measurable_of_tendsto_metrizable on the truncations at n : ℕ (pointwise eventually equal). Sceptic's warning: the absolute Kronecker factor is NOT characteristic for 3-term averages, and uniform Wiener–Wintner fails, without ergodicity. Counterexample f(t,x) = (t, x+t) on [0,1]×T: b = cos 2πx is weakly mixing (Riemann–Lebesgue in t), yet ∫ b·(cos 4πx)∘fⁿ·b∘f²ⁿ = 1/4 for all n. Use the relative Kronecker factor over the invariant σ-algebra in the non-ergodic case.