# T-invariant sub-sigma algebra

A $T$-invariant sub-sigma algebra of $\mathcal{B}$ where $(X,\mathcal{B},\mu,T)$ is a measure-preserving system is a collection $\mathcal{A}$ of elements of $\mathcal{B}$ that forms a sigma-algebra itself and satisfies $T^{-1}\mathcal{A} = \mathcal{A}$ mod $\mu$, meaning $\forall A \in \mathcal{A} \hspace{1mm} \exists B \in \mathcal{A}$ such that $\mu(T^{-1}A\Delta B) = 0$.

A problem I am working on discusses the smallest T-invariant sub-sigma algebra satisfying a certain property. The existence of such a thing presumably is based on the fact that the arbitrary intersection of $T$-invariant sub-sigma algebras is itself a sub-sigma algebra. This fact is clear to me if the definition were instead that $T^{-1}\mathcal{A} = \mathcal{A}$, without the mod $\mu$. However, since the intersection might be over an uncountable index, I do not see the proof or reason why this fact should be true with the “mod $\mu$” present.

Any help is appreciated.