Question

I recently learned, through my Topology coursework, that for any topological space $\mathbb{X}$, subset $A \subset \mathbb{X}$ and any quotient map $p : \mathbb{X} \rightarrow A$ that $$p^{-1}(\cup_{\alpha \in \Lambda}\mathcal{U_\alpha}) = \cup_{\alpha \in \Lambda}{(p^{-1}(\mathcal{U_\alpha}))}$$ $$p^{-1}(\cap_{\alpha \in \Lambda}\mathcal{U_\alpha}) = \cap_{\alpha \in \Lambda}{(p^{-1}(\mathcal{U_\alpha}))}$$ for indexed collections $\Lambda$ of open sets $\mathcal{U}$ in the quotient topology on $A$. It is true that similar facts hold for closed sets. So there is a homomorphism over set union and intersection operations. What is especially interesting to me is how broadly this homomorphism applies - it is respected for any surjective function $p$.

I am looking for similar examples of homomorphisms. These homomorphisms do not need to be over 'usual' ring operations. The best examples are homomorphisms which are true for large classes of functions.

Answer 1

A similar phenomenon occurs in measure theory. Consider subsets of some fixed set $\Omega$. A collection $R$ of subsets is called a ring if

$\emptyset \in R$,

$A,B\in R \Rightarrow A\backslash B \in R$, and

$A,B\in R \Rightarrow A\cup B \in R$.

A ring is called an algebra if $\Omega \in R$ also (this gives all complements, not just relative complements). You also get intersections because $A\cap B = A\backslash (A\backslash B)$. If you've ever heard the term $\sigma$-algebra, that is also an algebra, but closed under countable union.

Now for the "homomorphisms." A premeasure is any function $\mu$ from a ring $R$ to the non-negative real numbers with $\mu(\emptyset)=0$. A premeasure is called a content if for every sequence $(A_n)$ of pairwise disjoint sets of $R$,

$$\mu(\cup_{i=1}^n A_i) = \sum_{i=1}^n{\mu(A_i)}$$

From my reading of your post, this seems to be exactly the property of ``being a homomorphism'' you are looking for. If I'm mistaken, please let me know and I can probably come up with a different example. Note: if $R$ is a $\sigma$-algebra, then a premeasure is called a measure, which is the start of measure theory.

A lot of people seem to struggle with measure theory the first time they take it (I know I did!), and I think one of the reasons is that $\sigma$-algebras are often unmotivated and introduced without examples (e.g. in Big Rudin). I prefer the treatment above, which I learned from a great book called Elements of Measure Theory by Heinz Bauer.