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Let $\mu$ be a $\sigma$-finite measure on a measure space $(\mathbb{R}^d,\Sigma)$. Can $L^1_{\mu,loc}$ be represented as an injective-limit in the category of LCS (locally convex spaces and continuous linear maps) of the form: $$ \injlim L^{p_n}_{\mu_n} = L^1_{\mu,loc} $$ for some finite-measures $\mu_n$ on $(\mathbb{R}^d,\Sigma)$ and for some $p_n \in [1,\infty)$;

with the additional constraint that:

The resulting universal maps $\iota^n_{\infty}:L^1_{\mu_n}\rightarrow L^1_{\mu,loc}$ are the inclusions (ie: $f \mapsto f$)?

This is a refinement of this post.

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    $\begingroup$ What do you mean by the injective-limit? Here is my suggestion, for the case of Lebesgue measure. We cover the space by balls of radius $n$ and denote by $\mu_n$ its corresponding restrictions. The the Banach spaces $(E_n)$ form both an inductive and a projective system, where these are the corresponding $L^1$-spaces. The projective limit is naturally identifiable with the Fréchet space of (equivalent classes of) locally integrable functiins. The inductive limit is the strict $LF$-space of integrable functions with compact support. I am not sure which one you mean—hope this helps. $\endgroup$
    – user131781
    Commented Jan 14, 2020 at 18:31
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    $\begingroup$ As an additional remark, these are not just any old inductive or projective limits but have partitions of unity in the sense of de Wilde which makes them particularly tractable. $\endgroup$
    – user131781
    Commented Jan 14, 2020 at 18:35
  • $\begingroup$ Grubb treats these LF-spaces in the case where $\mu$ is Lebesgue and $d=1$, can the analogous construction still be identified with $L^p_{\mu,comp}(\mathbb{R}^d;\mathbb{R}^D)$ in general (ie for $\mathbb{R}^D$-valued functions with respect to any $\sigma$-finite measure)? That is, do you have a reference? $\endgroup$
    – ABIM
    Commented Jan 14, 2020 at 19:19
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    $\begingroup$ Again, do you mean inductive or projective limits (I am not familiar with the term „injective limit“)? As I wrote, the projective limit is a Fréchet space, the inductive limit an $LF$- ( even $LB$-) space. The same holds for any value of $p$ and for vector-valued (even Banach space valued) functions. I haven‘t really thought about the case of a general measure but imagine it will hold, say, for a locally finite measure on a $\sigma$-compact, locally compact space. $\endgroup$
    – user131781
    Commented Jan 14, 2020 at 19:45
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    $\begingroup$ Again, I don‘t have an explicit reference but would emphasise that the inductive limi in the lcs sense gives the space of functions with compact support, not the locally integrable ones. The inductive limit in the sense of Banach spaces gives precisely $L^1$. Glad to have been of help. $\endgroup$
    – user131781
    Commented Jan 14, 2020 at 20:34

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