Question about Kronecker factor

Question

In her paper Ergodic methods in additive combinatorics, Bryna Kra said that the Kronecker factor $(Z_1, \mathcal{Z}_1, m, T)$ of $(X, \mathcal{X},\mu,T)$ is the sub-$\sigma$-algebra of $X$ spanned by the eigenfunctions. Where $(X, \mathcal{X},\mu,T)$ is a measure preserving dynamical system.

But eigenfuctions are in $L^2(X,\mu)$, how dose they span a sub-$\sigma$-algebra of $X$.

Thanks a lot.

Answer 1

The sub-σ-algebra of $X$ spanned by the eigenfunctions means the smallest sub-σ-algebra of $\mathcal{X}$ such that the eigenfunctions are measurable.