I am not a specialist in measure theory, so excuse me if this is simple.

Let $\mu$ be a finite measure on a set $X$ (for example, the Lebesgue measure on $[0,1]$). Integrable functions on $X$ can be defined as limits (in different senses, in particular, in some definitions as uniform limits, but this does not play an important role in what I want to ask) of summable step functions $$ f=\lim_{n\to\infty}\sum_{i=1}^\infty\lambda_i^n\cdot\chi_{A_i^n}, $$ where $\lambda_i^n\in{\mathbb C}$, $A_i^n\subseteq X$.

Is it possible to control the set $\{\lambda_i^n\cdot\chi_{A_i^n};\ i,n\in{\mathbb N}\}$ in this equation so that it would not be "too big"?

For example,

Suppose $f\in L_1(\mu)$ and $|f(x)|\le 1$, $x\in X$. Do there exist a sequence of measurable sets $A_i\subseteq X$ such that

1) the set of indicators $\{\chi_{A_i};\ i\in{\mathbb N}\}$ is totally bounded (or, what is the same, precompact) in $L_1(\mu)$, and

2) $f$ belongs to the closed absolutely convex hull of the set $\{\chi_{A_i};\ i\in{\mathbb N}\}$ in $L_1(\mu)$?