I want to understand whether the following holds for all integrable $\varphi\in \mathcal L^1(\Omega, \mathcal A, P):$
Let $A = \bigcup_{n=1}^\infty A_n$ be a disjoint union of measurable sets $A_n \in \mathcal A$. Then $$\int_A \varphi \,dP = \sum_{n\geq 1}\int_{A_n}\varphi \, dP.$$
Now the Proof I saw writes $$1_{A} = 1_{\bigcup_{n=1}^\infty A_n} = \sum_{n=1}^\infty 1_{A_n}$$ since the $A_n$ are disjoint. Then the interchange of integral and limit is justified by the monotone convergence theorem. But I think that this is not applicable for negative $\varphi$ since the partial sums then must not be increasing. Is the statement wrong in this form or is there an alternative way of proving it?
This is justified by the dominated convergence theorem, not the monotone convergence theorem. Just apply it to the partial sums $\sum_{n=1}^N1_{A_n}\varphi$, which are bounded in absolute value by the integrable function $|\varphi|$.