Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$.

**Definitions**:

A subset $E\subseteq X$ is called locally Borel if $F \cap E$ is Borel for every Borel set $F\subseteq X$ with $\mu(F) < \infty.$ The locally Borel subset $E$ is called locally null if $\mu(E \cap F)= 0$ for every $F\subseteq X$ with $\mu(F) < \infty$. We say that a property of the points of $X$ holds locally almost everywhere (= a.e.) if the set of points where the property fails is locally null. A function $f: X \to \mathbb{C}$ is called locally measurable if $f^{-1}(A)$ is locally Borel for every Borel subset $A \subseteq \mathbb{C}$, i.e. $f$ is measurable w.r.t. the larger $\sigma$-algebra of locally Borel sets on the domain.

**Construction** $1$: [As e.g. defined in Folland's book "A course in abstract harmonic analysis", p50.]

Define $\mathscr{L}_1^\infty$ to be the set of locally measurable functions $f: X \to \mathbb{C}$ with the following property: there exists $\alpha \in (0, \infty)$ such that $$\{x \in X: \lvert f(x)\rvert > \alpha\}$$ is a locally null set. The vector space $\mathscr{L}_1^\infty$ has the seminorm $$\lVert f\rVert_\infty:= \inf\{\alpha \ge 0: \{x \in X: \lvert f(x)\rvert> \alpha\} \text{ is locally null}\}.$$

Let $\mathscr{N}_1$ be the subspace of locally measurable functions that are $0$ locally a.e.. We then define the quotient vector space $$L_1^\infty:= \mathscr{L}_1^\infty/\mathscr{N}_1$$ on which the seminorm $\lVert\cdot\rVert_\infty$ becomes a norm.

The standard argument proves that this is a Banach space.

**Construction 2**: [As e.g. defined in "Abstract harmonic analysis" by Hewitt and Ross, p151, or as e.g. defined in "Measure theory" by Cohn, section 3.3.]

Let $\mathscr{L}_2^\infty$ be the set of bounded measurable functions (with the usual supremum norm) and let $\mathscr{N}_2$ be the set of measurable bounded functions that are $0$ locally almost everywhere. Then we define the quotient vector space $$L_2^\infty:= \mathscr{L}_2^\infty/\mathscr{N}_2$$ (together with the quotient norm).

* Question*: Do these two constructions give the same Banach space, i.e. is there a canonical isometric isomorphism $$\Phi: L_2^\infty \to L_1^\infty?$$

There is an obvious well-defined linear isometry $$\Phi: L_2^\infty \to L_1^\infty: f + \mathscr{N}_2 \mapsto f + \mathscr{N}_1.$$

However, is it surjective? I.e. can we choose a representative of an element of $L_1^\infty$ to be measurable, instead of only locally measurable?

There is a possibility that this doesn't need any topological assumptions and that the above is true for general measure spaces and measures.

**Context question**: I'm reading the book "A course in abstract harmonic analysis" by Folland. On p50, it is claimed that the canonical map $L^\infty_1 \to (L^1)^*$ is an isometric isomorphism, but for the surjectivity Folland refers to Hewitt and Ross, where the other definition is used, so I'm trying to unify them. In this context, it would be desirable that the map $\Phi$ above is surjective.

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