# Dual of essentially compactly supported functions on a hemi-compact Radon space

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 linear 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 $$L_c(X)$$?

• Do you really mean the category of locally convex spaces and continuous maps or rather the continuous and linear maps? Moreover, $X_n$ is probably a compact exhaustion of $X$, right? Jun 4, 2021 at 16:11
• Roughly, the dual of the LB-space is the projective limit of the duals. My guess is thus, that $L_c(X)'$ is the space of measurable function whose restrictions to all $X_n$ are $\mu$-a.e. bounded. Jun 4, 2021 at 16:14
• @JochenWengenroth I was especially wondering if it had an interpretation as some sort of (Radon?) measures... do you think so? Also yes to your first point. Jun 8, 2021 at 12:06
• If $X$ is compact, the dual of $L^1(X,\mu)$ is $L^\infty(X,\mu)$. In the hemi-compact case, the dual is the projective limit of $L^\infty(X_n,\mu)$. What else do you want? Somewhat artificially, you can identify $L^\infty(X,\mu)$ with a space of measures, assigning to each bounded function $f$ the measure with $\mu$-density $f$. Jun 9, 2021 at 16:52