$L^1_{\mu}$ as limit

Question

Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. Does there exist a countable set of finite measures $\{\mu_n\}_{n \in \mathbb{N}}$ on $(X,\Sigma)$ such that $L^1_{\mu}(\Sigma)$ can be written as the projective-limit in the category of LCS $$ L^1_{\mu}(\Sigma) = \projlim\, L^1_{\mu_n}(\Sigma), $$ for some suitable restriction maps $\pi_n^m:L^1_{\mu_m}(\Sigma) \rightarrow L^1_{\mu_n}(\Sigma)$.

Related: Can $L^1_{loc}$ be represented as colimit?

Answer 1

There is a strictly positive integrable function $f\in L^1_\mu(\Sigma)$, hence $\nu=f\cdot \mu$ is a finite measure and $\Phi:L^1_\nu(\Sigma)\to L^1_\mu(\Sigma)$, $g\mapsto gf$ is an isomorphism. In particular, $L^1_\mu(\Sigma)$ is isomorphic to a projective limit of $L^1_{\mu_n}(\Sigma)$ with finite measures (and if you define a projective limit in the categorial sense by universal properties isomorphic to and equal to a projective limit is the same).

Let me add a general remark: Projective limits in the category of locally convex spaces are extremely useful to represent general spaces by simpler ones, in particular Frechet spaces as countable projective limits of Banach spaces. Useful results usually require an additional property called reducedness, i.e., the connecting maps $\pi_m^n:X_m\to X_n$ have dense range for $m\ge n$ or variants of that, like the Mittag-Leffler condition for all $n$ there is $m\ge n$ such that for all $k\ge m$ we have $\pi_m^n(X_m)\subseteq \overline{\pi_k^n(X_k)}$, the abstract Mittag-Leffler theorem then implies that this also holds for $k=\infty$ where $\pi_\infty^n$ is the map from the projective limit into the $n$th step. Your question asks for a representation of a Banach space by other Banach spaces which is somewhat queer to the theory because the limit of a reduced spectrum of Banach spaces is again a Banach space only if for all but finitely many $n$ the linking maps $\pi_m^n$ are isomorphisms.