The following is (taken from) an exercise (2.1.7) from Einsiedler and Ward’s Ergodic Theory with a view towards number theory:
Let $\mathsf{X}=(X,\mathscr{B},\mu,T)$ be a measure preserving system. A sub-$\sigma$-algebra $\mathscr{A}\subset\mathscr{B}$ with $T^{-1}\mathscr{A}=\mathscr{A}$ modulo $\mu$ is called $T$-invariant sub-$\sigma$-algebra.
What does $T^{-1}\mathscr{A}=\mathscr{A}$ modulo $\mu$ mean? I’m thinking something along the lines of $\mu(T^{-1}A \Delta A)=0$ for all $A\in\mathscr{A}$, but I’m not sure (the book never defines this, or I must have missed it).