Let $c_n$ be a sequence of real numbers with $\sum c_n$ converging conditionally  but not absolutely. Suppose $\delta_n > 0$ is another sequence with $\delta_n \to 0$, and $\sum c_n \delta_n$ converging also conditionally but not absolutely.

Does there exist, for every $L^1$ function $f: [0, 1] \to \mathbb R$, a bijection $\gamma: \mathbb N \to \mathbb N$, and a sequence of measurable sets $A_n$ with $\mu(A_n) = \delta_n$ such that 

$\sum c_{\gamma(n)}1_{A_{\gamma(n)}} \to f$,


in $L^1$ and pointwise a.e?

*Note: Here $\mu$ denotes the Lebesgue measure.*