If $\mu$ is not a positive measure, we cannot say anything about the negative part $\mu^-$ with the help of $\mu_{sf}$: If $\mu$ is a signed-measure, use the Hahn-decomposition to conclude that $\mu_{sf}(E) = \mu_{sf}^+(E)$. Thus, the statement is already false, if there exist a measurable functions $f$ such that $$\int |f|^2 \mu^- = \infty, \ \ \ \text{but} \ \ \ \ \int |f|^2 \mu^+ < \infty.$$$$\int |f|^2 \mathop{d \mu^-} = \infty, \ \ \ \text{but} \ \ \ \ \int |f|^2 \mathop{d \mu^+} < \infty.$$ Assuming that $\mu$ is a positive measure, the statement is also falsetrue.
In my first answer I have ignored that we only talk about equivalence classes of functions. We may have functions (One counterexample is given$f \neq g$ $\mu$-almost everywhere with $f \in L^2(\mu)$ and $g \notin L^2(\mu)$, but in this case $f = g$ $\mu_{sf}$-almost everywhere, as we will see below.)
One simple example is $\Omega = \{1\}$, $\Sigma = P(\Omega)$, $\mu = \infty \cdot \delta_1$. Here $1_\Omega \notin L^2(\mu)$, but $\mu_{sf} =0$ and thus $1_\Omega =0$ $\mu_{sf}$-almost everywhere.)
We know that any integrable function is already located on a $\sigma$-finite measurable subset (that follows from the Chebyshev's inequality). Note that for any set $E \in \Sigma$ with finite measure, i.e. $\mu(E) < \infty$, we have $\mu_{sf}(E) = \mu(E)$, just by definition and monotonicity. Moreover, if $E$ is $\sigma$-finite (i.e. $E= \bigcup_{n=1}^\infty A_n$ with measurable $A_n$ satisfying $\mu(A_n) <\infty$), then also $\mu(E) = \mu_{sf}(E)$. (Proof: Taking $E_m = \bigcup_{n=1}^m A_n$ and using measure continuity, we get $\mu(E) \leq \mu_{sf}(E)$. On the other hand, we have for any measurable $A \subset E$ already $\mu(A) \leq \mu(E)$ by measure-monotonicity.)
That observation implies for any $f \in L^2(\mu)$ $$\int |f|^2 \, \rm{d} \mu = 2 \int_0^\infty x \, \mu(|f| > x) \, \rm{d} x = 2 \int_0^\infty x \, \mu_{sf}(|f| > x) \, \rm{d} x = \int |f|^2 \, \rm{d} \mu_{sf},$$$$\int |f|^2 \mathop{d \mu} = 2 \int_0^\infty x \, \mu(|f| > x) \mathop{dx} = 2 \int_0^\infty x \, \mu_{sf}(|f| > x) \mathop{dx} = \int |f|^2 \mathop{d \mu_{sf}},$$ i.e. the map is well-defined and injective. (Moreover, ifin the sense that for $\mu$ is$f=g$ $\sigma$$\mu$-finitealmost everywhere, we have also $\mu = \mu_{sf}$$f=g$ $\mu_{sf}$-almost everywhere) and the map is bijectiveinjective.)
This map is also surjective.
HoweverProof: First, let us add a proof of the map don't havefact that $\mu_{sf}$ is a measure.
- Of course, $\mu_{sf}(\emptyset)=0$.
- Now, let $(A_n)_{n \in \mathbb{N}} \subset \Sigma$ be disjoint sets. If $\mu_{sf}(A_n) = \infty$ for some $n \in \mathbb{N}$, then we see easily that also $\mu_{sf}(E) = \infty$ for $E:= \bigcup_{n=1}^\infty A_n$. Thus, we can suppose that $\mu_{sf}(A_n) <\infty$ for all $n \in \mathbb{N}$. Taking $B_n \subset A_n$ with $\mu_{sf}(A_n) \leq \mu(B_n) + \frac{\varepsilon}{2^n}$ shows that $$\sum_{k=1}^n \mu_{sf}(A_k) \leq \sum_{k=1}^n (\mu(B_k)+\frac{\varepsilon}{2^k}) \leq \mu(\cup_{k=1}^n B_k) + \varepsilon \leq \mu_{sf}(E) + \varepsilon.$$ Thus $\sum_{k=1}^\infty \mu_{sf}(A_k) \leq \mu_{sf}(E)$. On the other hand, we find for any $A \subset E$ with finite measure that $\mu(A) = \sum_{n=1}^\infty \mu(A \cap A_n) \leq \sum_{k=1}^\infty \mu_{sf}(A_k).$
In order to beprove that this map is surjective, first assume that $g = 1_A \in L^2(\mu_{sf})$, i. Depending on thee. $\sigma$-algebra$\mu_{sf}(A) <\infty$. In this case, we may have not enough measurable sets with finite-measure to cover 'large sets' which are notcan take $\sigma$-finite$A_n \subset A$ with w. One really simple example: Letl.o.g. $\Sigma = \{ \emptyset, A, A^c, \Omega\}$$A_n \uparrow$, where $A \neq \Omega,\emptyset$,$\mu(A_n) < \infty$ and define $$\mu(B) = \begin{cases} 0 & \text{if } B = \emptyset \text{ or } B = A^c \\ \infty & \text{else} \end{cases}.$$ This is a measure on $(\Omega,\Sigma)$ such that$\mu(A_n) \uparrow \mu_{sf}(A)$. Define $1_A \notin L^2(\mu)$$B= \bigcup_{n=1}^\infty A_n$, becausethen $\mu(B) = \mu_{sf}(A)$. (Note that it can happen that $\mu(A) = \infty$ as in the previous example.) Because of $\mu(B) = \mu_{sf}(B)$, butwe get $\mu_{sf} =0$$\mu_{sf}(A \setminus B) =0$ and hencethus $1_A \in L^2(\mu_{sf})$$F(1_B)=1_A$, where $F \colon L^2(\mu) \rightarrow L^2(\mu_{sf})$ denotes the identity map.
Comment: I don't think this questionNow any $g \in L^2(\mu_{sf})$ can be approximated by simple functions $g_n$. Moreover, by the previous step, we find simple functions $f_n \in L^2(\mu)$ with $F(f_n) =g_n$. Using that $F$ is researchan isometry, we see that $(f_n)_{n \in \mathbb{N}}$ is a Cauchy-levelsequence in $L^2(\mu)$, say, with limes $f \in L^2(\mu)$. One can check easily that $F(f) =g$.