Does this norm on $L^1$ have a name? We can define a norm, $\| \cdot \|$ on $L^1(X, \mathcal{A}, \mu)$ by
$$\|f\| = \sup_{A \in \mathcal{A}} \bigg| \int_A f d\mu \bigg|$$
I found this norm referenced in a stats paper (they didnt say it was a norm) and was curious if it had a name or if it was related to some other concept like total variation or something.
 A: For real-valued functions $\in L^1(X,\mu)$ it is
$$
\frac{\|f\|_{L^1}+|\int_X fd\mu|}2$$ so not very interesting. No idea for complex-valued functions.
A: For complex-valued function, I don't think that there is an equation relating it to the $L^1$ norm, like the one @reuns provided for real-valued functions. However, as @David C. Ullrich pointed out in the comments, we can still show that $\| \cdot \|$ is equivalent to $\|\cdot \|_{L^1}$, and more precisely that :
$$\forall f\in L^1_{\mathbb C}(X,\mu), \quad \frac1{4\sqrt{2}}\|f\|_{L^1}\leq \|f\|\leq \|f\|_{L^1}$$
The rightmost inequality is immediate. Let $f\in L^1_{\mathbb C}(X,\mu)$.
For $k \in \{0,1,2,3\}$, let :
$$A_k = \left\{x \in X \middle| f(x) \neq 0 \text { and } \arg(f(x)) \in \left[k\frac\pi 2,(k+1)\frac\pi 2\right]\right\}$$
and $B = \{x \in X|f(x) = 0\}$.
Then, on $A_k$, we have :
$$|f|\leq \sqrt{2}\Re \left[e^{-i(k+1/2)\pi/4}f\right]$$
Therefore :
\begin{align}
\| f\|_{L_1} &= \int_X |f|\\
&= \sum_{k=0}^3 \int_{A_k}|f| \\
& \leq \sqrt{2}\sum_{k=0}^3 \Re \left[ e^{-i(k+1/2)\pi/4}\int_{A_k}f\right] \\
&\leq \sqrt{2}\sum_{k=0}^3 \left|\int_{A_k}f\right| \\
&\leq 4\sqrt{2}\|f\|
\end{align}
