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Under what conditions do the following statements hold?

$$\Re \left(\int f(x_i)\,dV\right)=\int\Re(f(x_i))\, dV$$

$$\Im \left(\int f(x_i)\,dV\right)=\int\Im(f(x_i)) \,dV$$

Where $x_i\in\mathbb{R}^n$ is the $n$-tuple of coordinates, $f(x_i):\mathbb{R}^n\to\mathbb{C}$, and $dV$ is a differential volume element in $\mathbb{R}^n$. I can see how the real and imaginary part "functions" tunnel through finite sums, but I don't know for sure how to treat the infinite sum and integral case.

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The integral operator is linear, so $$ \int f\,dV = \int (\Re f + i \Im f)\,dV = \int \Re f\,dV + i \int \Im f\,dV $$ Since you know that $\Re$ and $\Im$ teleport through finite-sums, they tunnel through Riemann integration which is defined to be some limit of finite-sums. Integrals defined in terms of measure are similar.

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