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 and **LCS** is the category of locally convex spaces and continuous linear maps.

*Related*: $L^1_{\mu}$ as limit

inductivespectrum of spaces and hence aninductivelimit (which is the same as a colimit). The corresponding topology in TOP (and also in the category LCS of locally convex spaces) is much finer than the Frechet topology of $L^1_{loc}$ (which is the reverse orprojectivelimit in LCS with respect to the restriction mappings). $\endgroup$setL^1_loc, so the colimit was assumed to be in the category of sets. Now that the question was retroactively edited, my answer no longer makes sense. $\endgroup$2more comments