Let $X$ be a hemicompact Radon space 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 maps.

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

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 $X$. Is there an analogous interpretation of $L_c(X)$?