Adding integrals with different domains

Question

Suppose I have two integrals

$$ \int_{\Omega_1} f \, \, d \eta$$ and $$ \int_{\Omega_2} g \, \, d \eta$$ how would I define the sum of these two integrals? Is it possible? I want something of the form

$$ \int_{\Omega_1 \cup \Omega_2} f + g \, \, d \eta$$

What is the best way to define such an integral?

Answer 1

Let $\chi_1$ and $\chi_2$ denote the characteristic functions on $\Omega_1$ and $\Omega_2$, respectively. (In other words, $\chi_k(x)=1$ if $x \in \Omega_k$ and is $0$ otherwise.) Then you can write the desired sum as $$ \int_{\Omega_1 \cup \Omega_2} f \chi_1 + g \chi_2 \, \, d \eta$$