In Bishop and Cheng's paper, "Constructive Measure Theory", page 18, they define the characteristic function of a complemented set $A=(A_1,A_2)$ by the function $\chi : A_1\cup A_2 \to \mathbb{R}$, such as $\chi(x)$ is 1 if $x\in A_1$ and 0 if $x\in A_2$.
As a complemented set, $A_1$ and $A_2$ are disjoint, so this looks like a law of excluded middle : either $x\in A_1$ or $x\notin A_1$. But that law is not constructive.
Or is it that in a constructive setting, a union $x \in A_1\cup A_2$ is assumed to be either a proof that $x\in A_1$, or a proof that $x\in A_2$, so that we are allowed to make the previous definition of $\chi$ by case analysis ?
The latter option is correct: a union is taken to be tagged in this way. $A \cup B$ can be defined with two constructors: $\mathrm{inl} : A \to A \cup B$, and $\mathrm{inr} : B \to A \cup B$.