Consider the ON-sequence $\{\varphi_{k}\}_{k}\in L^{2}(\mathbb{R})$ and let $I_{k}\in \mathrm{Bernoulli}(\lambda_{k}),\;\sum_{k=1}^\infty \lambda_{k}<\infty$ $ \{I_k\} $ are all independent. Also let $A$ be a bounded intervall in $\mathbb{R}$ Is it allowed to change order of integration in the following manner? $\mathbb{E}\left[\intop_A \intop_{A^c}\sum_{k=1}^\infty \sum_{j=1}^\infty I_k I_j \varphi_k(x)\overline{\varphi_j(x)}\overline{\varphi_k(y)}\varphi_j(y) \, dx \, dy\right]=$ $\intop_{A}\intop_{A^{c}}\sum\limits _{k=1}^{\infty}\sum\limits _{j=1}^{\infty}\mathbb{E}\left[I_{k}I_{j}\right]\varphi_{k}(x)\overline{\varphi_{j}(x)}\overline{\varphi_{k}(y)}\varphi_{j}(y)dxdy$
This is how far i have come on my own: I would like to invoke Borell Cantelli lemma, which says that the sum in the integral is finite a.s.. However for each $\omega\in\Omega$ outside a zero-set the number of terms will depend on $\omega$ so there is no single function that bounds the integrand for almost all $\omega$ and there for we may not move the expectation inside the integral.