Let us consider the space $L^1(0,1;\mathbb{R})$ of real-valued, Lebesgue integrable functions defined on the interval $(0,1)$ (where we only distinguish functions which are not equal almost everywhere). Let us define the function $S \colon L^1(0,1;\mathbb{R}) \to 2^{(0,1)}$ given by the formula: \begin{equation} \forall f \in L^1(0,1;\mathbb{R}) \quad S(f) = \{ A \subseteq (0,1) \mid \int_A |f| \, \mathrm{d}x = \frac{1}{2} \lVert f \rVert_{L^1} \}. \end{equation}

Is there any simple way to characterize all $g \in L^1(0,1;\mathbb{R})$ such that for previously given $f$ we have $S(f) = S(g)$?

It is easy to see that we can multiply $f$ by any $h \in L^\infty(0,1;\mathbb{R})$ such that $|h|=c$ almost everywhere, where $c \ge0$, and we would have $S(fh ) = S(f)$, however, I am not sure whether this exhausts all possiblities.