Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be Lebesgue integrable and vanishing outside $[0,1)^d$. Moreover, let $k\in\mathbb N$ and $x_1,\ldots,x_k\in[0,1)^d$. Let $w_1,\ldots,w_k\ge0$ with $\sum_{i=1}^kw_i=1$ and suppose we want to approximate $$I:=\int f(x)\:{\rm d}x$$ by $$\tilde I:=\sum_{i=1}^kw_if(x_i).$$ If we write $$\sigma(x):=\sum_{i=1}^kw_i\delta(x-x_i)\;\;\;\text{for }x\in\mathbb R^d$$ and denote the Fourier transform of a function $g$ by $\hat g$, we see that $$\tilde I=\int\hat\sigma(\omega)\hat f(\omega)\:{\rm d}\omega.$$
Question: Assume now that the $x_i$ are random variables on a probability space. In equation 17 of this paper it is claimed that $$\operatorname E\left[\left|\int\hat\sigma(\omega)\hat f(-\omega)\:{\rm d}\omega\right|^2\right]=\int\operatorname E\left[\left|\hat\sigma(\omega)\right|^2\left|\hat f(-\omega)\right|^2\right]\:{\rm d}\omega\tag1.$$ However, I have no idea how they obtain $(1)$. They claim this would follow by "expanding the square and the expected value" and noting that "$\hat\sigma$ is random". The latter is clearly nonsensically (if you ask me), but even when I assume that the $x_i$ are independent, I' not able to derive $(1)$.
So, what do we need to do? Since the same equation can be found in equation 6 of this paper, the claim should be true.