Subspace of $L^1(\Omega)$ closed

Question

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ two sub-sigma-algebras and $f$ a $\mathcal{B}$-measurable function. I want to show, that the subspace $$\mathcal{L}:=\{\langle\phi,f\rangle : \phi\text{ measurable with respect to }\mathcal{A}\}\cap L^1{\Omega}$$ is a closed subspace of $L^1$. Unfortunately I don't know how to start here. I'm thankful for every kind of help :)