Let $I:= [0,1]$ and let $L^1(I)$ be the space of integrable functions $f \colon I \to \mathbb{R}$, i.e. $f \in L^1(I)$ if and only if $\int_I|f| dm< \infty$. For $f \in L^1(I)$, consider the two norms :
$$|f|_{L^1(I)}:= \int_I |f| dm $$ and $$|f|_{TV}:= \sup_{A,B \subset I, \,A \cup B = \mathbb{S}^1, A\cap B= \emptyset} \left|\int_I \left(1_A-1_B\right)f\right|,$$
where $1_A,1_B$ denote the indicator functions resp. in $A$ and $B$. This is the total variation norm of $f$ seeing $f$ as a countably additive signed measure. My question is the following:
Are these two norm equivalents in $L^1(I)$, i.e. can we find constants $C,c>0$ such that
$$C|f|_{TV} \ge |f|_{L^1} \ge c|f|_{TV} $$,
for all $f \in L^1(I)$