For a probability space $(\Omega, \mathscr{F}, \mathbb{P})$ with sub-$\sigma$-algebra $\mathscr{G}\subseteq \mathscr{F}$, we have the orthogonal $L^2$-decomposition
\begin{equation} L^2(\Omega, \mathscr{F}, \mathbb{P}) \,=\, M \oplus M^{\bot} \qquad\text{for}\quad M:=L^2(\Omega, \mathscr{G}, \mathbb{P}). \end{equation}
Are there conditions on $\mathscr{G}$ to guarantee that $M^{\bot}= L^2(\Omega, \mathscr{G}_{\bot}, \mathbb{P})$ for some sub-$\sigma$-algebra $\mathscr{G}_{\bot} \subseteq \mathscr{F}$?
No complement (even non-orthogonal) of that $L^2$ space can be the $L^2$-space of another sub-$\sigma$-algebra for a finite measure (even a different one, as long as the underlying measured space is the same) due to constant functions being shared by both of them.