Does $\sum_{n=1}^N \int f_n \mathrm d\mu = \int \sum_{i=1}^N f_n \mathrm d\mu$ hold if $\int f_1 \mathrm d\mu = \infty$? Assume $f_1,\ldots,f_N:\mathbb R \to \mathbb R$. If $\int |f_n| \mathrm d\mu < \infty$ for all $n=1, \ldots, N$, then we always have $$\sum_{n=1}^N \int f_n \mathrm d\mu =  \int \sum_{i=1}^N f_n \mathrm d\mu.$$
Now we assume $f_1,\ldots,f_N:\mathbb R \to [0, \infty]$ with $\int f_1 \mathrm d\mu = \infty$. I would like to ask if $$\sum_{n=1}^N \int f_n \mathrm d\mu =  \int \sum_{i=1}^N f_n \mathrm d\mu$$ still holds. Thank you so much for your explanation!

Yes, and for trivial reasons. We have
$$\sum_{n=1}^N \int_X f_n d\mu \ge \int_X f_1 d \mu = \infty$$
so
$$\sum_{n=1}^N \int_X f_n d \mu = \infty.$$
Next, by monoticity of integral, $$\int_X \sum_{n=1}^N f_n d \mu \ge \int_X f_1 d \mu = \infty$$
so we have
$$\int_X \sum_{n=1}^N f_n d \mu = \infty$$
and both sides are $\infty$.
