I would like to show that $\int_{E\cup F}f=\int_E f+\int _F f$, where $E\cap F=\emptyset$ and $E,F$ are Lebesgue measurable sets.
Attempt:
First I tried to show that in general I can write $\int (f+g)=\int f+\int g$ like so:
Proof.
Suppose that there are two sequences $(\varphi_n)$ and $(\psi_n)$ such that $\lim_{n\to\infty}\varphi_n=f$ and $\lim_{n\to\infty}\psi_n=g$, where these sequences are made up of non-negative, integrable, simple functions. Then applying Lebesgue's Monotone Convergence Theorem (that is, if we have $f_i$ non-negative and measurable: $f_1\leq\ldots\leq f_n\leq\ldots$, then $\lim_{n\to\infty}\int f_n=\int\lim_{n\to\infty}f_n$), we can get the following:
\begin{align} &\int (f+g)\\ &=\lim_{n\to\infty} \int(\varphi_n+\psi_n)\\ &=\lim_{n\to\infty}\int \varphi_n+\lim_{n\to\infty}\int \psi_n\\ &=\int f+\int g \end{align}
So I (think) I've proved that $\int_T (f+g)=\int_T f+\int_T g$, but this is for a common domain $T$. I'm not sure how to extend this result to split the integral with different domains. (Perhaps there is an easier way to show this instead?)
Any help would be appreciated. Thanks.