So this question lingered in my mind for quite some time: Is it possible to develop "algebraic measure theory" analogous to algebraic topology, and can it be useful?
What I mean is the following: Let $(X,\Sigma)$ be a measurable space, and fix $x_0\in X$. Let $L_{x_0}:=\{\gamma:[0,1]\to X \text{ measurable }|\ \gamma(0)=\gamma(1)=x_0\}$ be the set of measurable loops based at $x_0$ (where $[0,1]$ is equipped with the Borel $\sigma$-algebra). Two loops $\gamma_0,\gamma_1\in L_{x_0}$ are said to be 'measure equivalent' if there exists a measurable function $H:[0,1]\times [0,1]\to X$ such that $H(-,i)=\gamma_i$ for $i\in\{0,1\}$, and $H(i,t)=x_0$ for all $i\in\{0,1\}$ and $t\in[0,1]$. We denote measure equivalence by $\sim$. Then with the same proofs as in the beginning of algebraic topology, one can equip $L_{x_0}/\sim$ with a group structure via concatenating loops, i.e. if $[\gamma_0],[\gamma_1]\in L_{x_0}/\sim$ are two classes of loops and $\gamma_0 *\gamma_1$ denotes the concatenation of $\gamma_1$ and $\gamma_2$, then the operation $[\gamma_0]\cdot[\gamma_1]:=[\gamma_0 *\gamma_1]$ defines a group structure on $L_{x_0}/\sim$.
Are there situations where $L_{x_0}/\sim$ can be computed and it isn't the trivial group? To what extent can one carry out the same proofs as in algebraic topology? And has this been investigated at all?