Let $(X, M, \mu)$ be a measure space and $F:M \to \mathbb{R}$ be an absolutely continuous countably additive function(i.e., $F(\cup_n^\infty E_n) = \sum_n^\infty F(E_n)$ for disjoint $E_n \in M$).
There is a Radon-Nikodym derivative $f$ and $F(E)=\int_E f d\mu$.
Does the following hold?
$$f(x)=\lim_{\mu(E)\to 0, x\in E \in M} \frac{F(E)}{\mu(E)} (a.e.)$$
What I tried:
- It is enough to prove for the case $F\geq0$. Let $\mathcal{F}$ be the set $\{ f\geq0| \forall E\in M. \int_Ef\mu\leq F(E)\}$ and $\alpha = \sup_{f \in \mathcal{F}}\int_Xf d\mu$. If I select functions $f_n \in \mathcal{F}$ such that $\alpha = \lim_{n \to \infty} \int_X f_nd\mu$, $\sup_{n\geq 1}f_n$ is a Radon-Nikodym derivative of $F$. Hense I want to find $f_n$ in the form $f_n(x)=\frac{F(E_x)}{\mu(E_x)}$ for some $E_x \in M$ satisfying $x\in E_x$, $\mu(E_x) < \frac{1}{n}$ and $f_n \in \mathcal{F}$. Next, I want to prove $\alpha = \lim_{n \to \infty} \int_X f_nd\mu$ and $\sup_{n\geq 1} f_n = \lim_{\mu(E)\to 0} \frac{F(E)}{\mu(E)}(a.e.)$. However, I do not know how to construct such measurable $f_n$.
- Let $f$ be a Radon-Nikodym derivative of $F$ and $S$ be the set $\{x \in X | f(x) \neq \lim_{\mu(E)\to 0} \frac{F(E)}{\mu(E)}\}$. $S=S_0 \cup \cup_{n \in \mathbb{N}}S_n$, where $S_0$ is the set of $x$ where $\lim_{\mu(E)\to 0, x \in E} \frac{F(E)}{\mu(E)}$ does not converge and $S_n$ are the sets which the limit converges and $\left|\lim_{\mu(E)\to 0, x \in E} \frac{F(E)}{\mu(E)}- f(x)\right|> \frac{1}{n}$. I want to prove $\mu(S_0)=0$. Then, if $\mu(S)\neq 0$, there is $S_n$ such that $\mu(S_n)\neq 0$ and I want to derive a contradiction using $S_n$. However, I cannot prove them.
Best Answer
Does the following hold? $$ f(x)=\lim_{\mu(E)\to 0, x\in E \in M} \frac{F(E)}{\mu(E)} \text{ (a.e.)} $$ In general, no. In most cases that limit fails to exist.
When something like it does hold, it is known as a derivation theorem. The classical reference on derivation theorems is:
Hayes, C. A.; Pauc, C. Y., Derivation and martingales, Ergebnisse der Mathematik und ihrer Grenzgebiete. 49. Berlin-Heidelberg-New York: Springer-Verlag. VII, 203 p. (1970). ZBL0192.40604.
A simple example is where $\mu$ is Lebesgue measure in the real line. A condition which does work:
replace "$x \in E \in M$" with "$E$ is an interval with midpoint $x$".