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as part of a proof in a paper i have statement, i cannot figure out how to proof:

Assume $(c_k)_{k\in \mathbb{N}}$ is a sequence of nonnegative random variables and $g: (-1,1] \to \mathbb{R}$ is a function such that
$f(x,\omega):= \sum_{k=0}^{\infty}c_k(\omega)x^k$
$\mathbb{E}(f(x,\cdot) \leq g(x)$
for all $x\in (-1,1]$.
Then
$\mathbb{E}(f(x,\cdot)= \sum_{k=0}^{\infty}\mathbb{E}(c_k)(\omega)x^k $
for all $x\in (-1,1]$.

The only explanation in the paper is ...apply Fubinin's theorem..., but i can't find one which can be applied here.

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For $x\ge0$, Fubini-Tonelli and for $x<0$, Fubini-Lebesgue.

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    $\begingroup$ Thanks. I solved it now with monotone convergence (for $x \geq 0 $ ) theorem and dominated convergence theorem ($ x < 0$). $\endgroup$
    – jekodo
    Commented Sep 29, 2019 at 6:45

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