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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.

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2 Answers

up vote 17 down vote accepted

This does exist, and has a nice explicit description. Treating the sets $A_i$, for convenience, as disjoint subsets of $A$, take a subset $S \subseteq A$ to be measurable exactly if $S \cap A_i$ is a measurable subset of $A_i$, for each $i$. The proof that this is a sigma-algebra making each $\psi_i$ measurable, and is the finest such, is reasonably straightforward.

From a categorical point of view, one can find this description by saying: if such a σ-algebra exists, one would hope that it should make $A$ a coproduct of the $A_i$'s in the category of measurable spaces.

But measurable subsets $S \subseteq A$ must correspond to measurable functions $f \colon A \to 2$ (this holds for any measurable space); hence, by the universal property of $A$, to families of functions $f_i \colon A_i \to 2$; hence to families of measurable sets $S_i \subseteq A_i$; thinking about naturality shows that this correspondence has to be via $S \mapsto (A \cap S_i)_{i \in I}$, and so leads to the description above. (And one can check then that this does indeed give a coproduct.)

One can see this as talking about a duality between the category of measurable spaces and a suitable category of lattices: the coproduct as spaces corresponds to the product of the lattices of measurable subsets.

I have no references, I’m afraid, since I don’t know of any categorically-minded treatments of measure theory. But any such book would surely include this construction; I’m hopeful that there’s one out there that I don’t know of?

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Sorry for a silly correction, but $A \mapsto (A \cap S_i)_{i \in I}$ should be $S \mapsto (S \cap A_i)_{i \in I}$ (which is obviously what you meant). –  L Spice Mar 30 '11 at 18:57
@L Spice: thanks; fixed! –  Peter LeFanu Lumsdaine Apr 5 '11 at 17:02
@PeterLeFanuLumsdaine Do you know why this construction you mention is more commonly used than another reasonable seeming alternative: The sigma algebra generated by the measurable sets of each of the $A_i$ This would sort of parallel the "countable cocountable" construction as opposed to yours which is the analog of "power set." More rigorously, your disjoint union of uncountably many 1 point spaces would give power set sigma algebra, but mine would give countable cocountable. –  Jeff May 20 at 21:14
Mine has the unfortunate consequence that you cannot piece together measurable functions from each piece, but you normally can't do that anyway. –  Jeff May 20 at 21:14
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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.

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