I was self studying the book on Stochastic integration and differential equations by Protter and in Theorem 52 Chapter 1, I do not understand the following

If $\mu$ and $\nu$ are two Borel measures on $\mathbb{R}_+$ with $\mu$ absolutely continuous w.r.t $\nu$ and if ones sets $$\alpha(s,t)=\frac{\mu((s,t])}{\nu((s,t])}, \text{if } \nu((s,t])>0$$ and 0 otherwise and then define

$h(t)=\liminf_{s \uparrow t} \alpha (s,t)$ and $h(t)=\limsup_{t \downarrow s} \alpha (s,t)$ then both $h,k$ are borel measurable and moreover they are versions of the Radon Nikodym derivative i.e $$d\mu=hd\nu \text{ and } d\mu=k d \nu$$

It is clear that since $\alpha$ is borel measurable its $\limsup$ is borel measurable. But I do not understand why do they have to be versions of the Radon Nikodym derivative .


This statement is called Lebesgue differentiation theorem.


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