Let $L^1\_{loc}$ denote the set of all functions from $\mathbb{R}$ to itself which are locally integrable. For every infinite compact subset $K\subseteq \mathbb{R}$, let $L^1\_{m\_K}$ denote the space of Lebesgue measurable functions supported on $K$. 


Clearly the collection $\mathcal{K}$ of all such compact subsets of $\mathbb{R}$ form a poset wrt inclusion $i^{K\_1}\_{K\_2}:K\_1\hookrightarrow K\_2$ if and only if $K\_1\subseteq K\_2$, for $K\_i \in \mathcal{K}$. Therefore, we may define the colimit
$$
\operatorname{colim}\_{\mathcal{K}} L^1\_{m\_K},
$$
in **Top**. 


How are $\operatorname{colim}\_{\mathcal{K}} L^1\_{m\_K}$ and $L^1\_{loc}$ related?


*Note/Edit*: **Top** is the category of [topological spaces and continuous maps](https://ncatlab.org/nlab/show/Top) and **LCS** is the category of [locally convex spaces and continuous linear maps.](https://ncatlab.org/nlab/show/locally+convex+topological+vector+space#properties) 


*Related*: [$L^1\_{\mu}$ as limit](https://mathoverflow.net/questions/350223/l1-mu-as-limit?noredirect=1#comment877508_350223)



Can $L^1_{loc}$ be represented as colimit?
Interpreting the question as Dmitri Pavlov I assume that $L^1\_K$ is the space of $L^1$-functions with support in $K$ so that we have an inductive spectrum of Banach spaces (identifying almost everywhere equal functions). The colimit (=inductive limit) in the category of locally convex spaces (i.e., the union endowed with the finest locally convex topology making the inclusions from all $L^1\_K$ continuous) is then a complete locally convex space (by a classical result of Dieudonne and Schwartz about *strict* countable inductive limits -- instead of all compact sets it is of course enough to consider a countable exhaustion). On the other hand it is dense in $L^1\_{loc}$ (the projective limit of all $L^1\_K$ with respect to the restrictions). Therefore, the inductive limit topology is *strictly* finer.
(There are several other ways to see this. For example, a countable inductive limit of normed spaces $X\_n$ such that $X\_n\neq X\_{n+1}$ is [never metrizable](https://math.stackexchange.com/questions/356085/inductive-limit-topology-and-first-countability).)


The limit topology in the category TOP is even finer than that in LCS.


If you consider uncountable colimits in LCS the situation is slightly different: As a Frechet spaces $L^1\_{loc}$ is *ultrabornological* and hence the inductive limit of Banach spaces, namely of all Banach spaces generated by absolutely convex closed bounded sets (generated means that you take the linear span endowed with the Minkowski functional). You can describe these spaces as *weighted* $L^1$-spaces $\{f\in L^1: \int |f|wd\mu<\infty\}$ with suitable weights.


