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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 $\sum |c_n| \delta_n = \infty$.

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.

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.

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

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.

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 $\sum |c_n| \delta_n = \infty$.

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.

Let $c_n, \delta_n$ be sequences of real numbers with $c_n, \delta_n \to 0$, with $\sum c_n, \sum \delta_n$ converging 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

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

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

Note: Here $\mu$ denotes the Lebesgue measure, and $\text{sgn}$ denotes the signum function.

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

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.