a measurable map is a measure-homomorphism? Is it true that a measurable map between two measure spaces, can be called a "measure-homomorphism"? 
The reason I think so, is because a homomorphism is a structure-preserving map, and a measurable map preserves the measure-structure. 
However, I'm not sure, because wikipedia says "homomorphism" applies to "algebraic structures", which are structures on operations on a set, but a measure on $A$ is not a structure on operations on $A$, but a structure on the power set of a set $A$.
 A: We can definitely consider the category of measure spaces with measurable maps, as they are closed to composition and all identity maps are measurable. 
Said that, we can calmly call them morphisms.
A: I think it is important to distinguish measurable spaces and measure spaces.
Recall that a measurable space is a pair $(X,\mathcal{B}(X))$ where $X$ is a set and $\mathcal{B}(X)\subseteq \mathcal{P}(X)$ is a $\sigma$-algebra. If we define a measurable arrow $\varphi:(X,\mathcal{B}(X))\to (Y,\mathcal{B}(Y))$ to be a function $\varphi: X\to Y$ with the property that $\mathcal{B}(\varphi):=\varphi^\ast:\mathcal{B}(Y)\to \mathcal{B}(X), T\mapsto \{s\in X| \varphi(s)\in T\}$, then we have a category (which I leave to you to verify). Observe that in standard books analysis books it is common to define a measurable function to be one that pulls open sets back to measurable sets, in which case the categorical framework breaks down (because in that situation compositions are jeopardized). Also observe that this definition is analogous to the definition of continuous functions (by replacing $\mathcal{B}:Set\to Set$ with $\mathcal{T}:Set \to Set$; as a further exercise one can establish these two as subfunctors of the powerset functor, and even specify the codomain of the first one to a much specific category, e.g. that of Boolean rings with $1$).
As for a measure space, we now have a triple $(X,\mathcal{B}(X),\mu)$, where the first two are as before and $\mu:\mathcal{B}(X)\to \mathbb{C}$ is a function that is countably additive with $\emptyset\mapsto 0$. If we were to take as arrows $(X,\mathcal{B}(X),\mu)\to (Y,\mathcal{B}(Y),\nu)$, we would let the measures go to waste. Thus not only will we require an arrow $\varphi: (X,\mathcal{B}(X),\mu)\to (Y,\mathcal{B}(Y),\nu)$ to be measurable, but also measure-preserving, i.e., that $\varphi_\ast(\mu):=\mu\circ \varphi^\ast=\nu$ (drawing the appropriate diagrams is on you). As a result we again get a category (again I am leaving it to you to verify this). Considering again the analogy of measurability versus continuity, this also hints at considering measure theory as some sort of quantitative topology (especially if you consider Borel $\sigma$-algebras together with Borel measures etc.).
The structure of measurability or measures is indeed very algebraic because essentially it is functional analytical: we can embed the $\sigma$-algebra of a space into the space of measurable functions (by $B\mapsto\chi_B$), and consider a measure to be a functional on this space (which starts out by first getting rid of the original space by $x\mapsto \delta_x=\operatorname{ev}_x$). Further, we could think of a measure $\mu$ as a way of quotienting out a $\sigma$-ideal (parametrized by $\mu$ as $\mu$-negligible sets) from the $\sigma$-algebra, which is precisely what we do to turn integrals into metrics ($\sigma$ means that we allow countable operations, which might be accounted for by some notion of "$\infty$-arity"). Indeed, looking at measure structures this way goes at least as back as von Neumann, consequently these ideas are well known in operator theory and dynamics/ ergodic theory. Halmos' Measure Theory book and Rohlin's classic paper "On The Fundamental Ideas Of Measure Theory" are testament to this.
That said, in more conventional areas, like, say probability theory, I highly doubt that this framework is common, because, as far as I can tell, category theory is not very adequate to do things like rate-of-convergence estimates (even though I personally find categorical framework useful even in probabilistic settings; for instance saying that a sequence of random variables $\{X_n\}_n\in L^0_b((\Omega,\mathbb{P}),\mathbb{C})$ is identically distributed is the same as saying ${X_n}_\ast(\mathbb{P})$ is the same measure on $\mathbb{C}$).
