Let $(\Omega, \mathcal{F}, P)$ be a probability measure space and $\mathcal{L}^1(\mathcal{G})$ =$\{$X $\mathcal{G}$-measurebale , $\int XdP<\infty$ $\}$ for $\mathcal{G}\subset\mathcal{F}$ an abritrary sub $\sigma$-Algebra.
$\circ$ Is $\mathcal{L}^1(\mathcal{G})$ modulo Nullsets with $|| \cdot ||_1$ a closed subset of $\mathcal{L}^1(\mathcal{F})$ / a Banach space?
$\circ$ Is there some other Subspace $\mathcal{S}$ such that $\mathcal{L}^1(\mathcal{G}) \oplus \mathcal{S} =\mathcal{L}^1(\mathcal{F})$ or even a $\mathcal{H}\subset \mathcal{F}$ with $\mathcal{L}^1(\mathcal{G}) \oplus \mathcal{L}^1(\mathcal{H}) =\mathcal{L}^1(\mathcal{F})$?