Let $X$ be a [hemicompact][1] [Radon space][2] and fix a $\sigma$-finite Radon measure $\mu$ on $X$.  Let $L(X_n)$ denote the subspace of $L_{\mu}^1(X)$ of "functions" which are $\mu$-essentially supported on $X$.  Let $L_c(X)$ denote the colimit (LB-space) of the direct system $\{L(X_n)\hookrightarrow L(X_m):\, n\leq m\}$ in the category of locally-convex spaces and continuous linear maps.  

**My question is**:  *Is there a concrete interpretation, possibly in terms of measures on $X$, of the continuous dual $L_c(X)'$?*

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**For comparison:**
I know that the continuous dual of $C_c(X)'$ (with its $LB$-topology; constructed in the analogous manner) [can be identified with the set of Radon measures on $X$][3].  Is there an analogous interpretation of $L_c(X)$?


  [1]: https://ncatlab.org/nlab/show/hemicompact+space
  [2]: https://encyclopediaofmath.org/wiki/Radon_measure
  [3]: https://mathoverflow.net/questions/220511/questions-on-topologies-on-space-of-radon-measures