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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 $L_c(X)$?

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  • $\begingroup$ 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? $\endgroup$ Jun 4, 2021 at 16:11
  • $\begingroup$ 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. $\endgroup$ Jun 4, 2021 at 16:14
  • $\begingroup$ @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. $\endgroup$ Jun 8, 2021 at 12:06
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    $\begingroup$ 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 $f$. $\endgroup$ Jun 9, 2021 at 16:52

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