I am confused between definitions from my text book and this website. This is how I see them defined now:
$\mathcal L^1({\mathbb{T}})= $ {$F:\mathbb{T}\rightarrow\mathbb{C}: F$ measurable and $\int_{\mathbb{T}}|F(\zeta)|d\lambda < \infty $}
$L^1({\mathbb{T}})$ is the quotient vector space $\mathcal L^1({\mathbb{T}})$/{$F\in \mathcal L^1({\mathbb{T}}) : \|F\|_1=0 $}
And they both have the same norm $\|F\|_1 = \int_{\mathbb{T}}|F(\zeta)|d\lambda$
Initially, I found definitions for $L^1$ which were indentical to $\mathcal L^1$. This may have been a problem of notation.