TITLE: Is there a "disjoint union" sigma algebra?
QUESTION [16 upvotes]: I'm looking for a measure-theoretic analogue to the disjoint union topology, or for work on the $\sigma$-algebra generated by canonical injections. More formally:
For an indexed family of sets $\{A_i\}_{i \in I}$, define $\psi_i : A_i \to A$, $a \mapsto (a,i)$ (the canonical injections), where $A = \bigcup_{i \in I} (A_i \times \{i\})$. Then the disjoint union $\sigma$-algebra $\mathcal{A}$ is the finest $\sigma$-algebra on $A$ such that for every $i \in I$, $\psi_i$ is $\mathcal{A}_i$-$\mathcal{A}$-measurable.
There are so many analogies between measure theory and topology that I've been surprised at how difficult it's been for me to find anything on this.
I'd appreciate references to any related ideas as well.

REPLY [10 votes]: David H. Fremlin's "Measure Theory", vol 2, 214K, gives this construction explicitly. He also proves some elementary properties, but unfortunately stops short of universal properties such as in Peter's insightful answer.