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.

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.
