I recently had the idea that maybe measure spaces could be viewed as sheaves, since they attach things, specifically real numbers, to sets...

But at least as far as I can tell, it doesn't quite work - if $(X,\Sigma,\mu)$ is a measure space and $X$ is also given the topology $\Sigma$, then we do get a **presheaf** $M:O(X)\rightarrow\mathbb{R}_{\geq0}$, where $\mathbb{R}_{\geq0}$ is the poset of non-negative real numbers given the structure of a category, and $M(U) = \mu(U)$, and $M(U \subseteq V)$ = the unique map "$\geq$" from $\mu(V)$ to $\mu(U)$, but this is not a sheaf because for an open cover {$U_i$} of an open set $U$, $\mu(U)$ is in general not equal to sup $\mu(U_i)$ (and sup is the product in $\mathbb{R}$).

First off, is the above reasoning correct? I'm still quite a beginner in category theory, but I've seen on Wikipedia something about a sheafification functor - does applying that yield anything meaningful/interesting? Does anyone have good references for measure theory from a category theory perspective, or neat examples of how this perspective is helpful?