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][1] and **LCS** is the category of [locally convex spaces and continuous linear maps.][2]  


  [1]: https://ncatlab.org/nlab/show/Top
  [2]: https://ncatlab.org/nlab/show/locally+convex+topological+vector+space#properties