I have not deep knowledge in measure theory, and I am wondering if one could help me with this question.
Based on what is defined in Wikipedia, It turns out that non-atomic measures actually have a continuum of values. It can be proved that if ${\displaystyle\mu }$ is a non-atomic measure and ${\displaystyle A}$ is a measurable set with ${\displaystyle \mu (A)>0,}$ then for any real number ${\displaystyle b}$ satisfying ${\displaystyle \mu (A)\geq b\geq 0}$ there exists a measurable subset ${\displaystyle B}$ of ${\displaystyle A}$ such that ${\displaystyle \mu (B)=b.}$
Here is the question:
Let $(\Omega,M,\mu)$ is an atomless probability space and $B(\Omega)$ be the space of all bounded random variable on $\Omega$, if $\nu:B(\Omega)\to R$ is a finite measure on $M$ wrt $\mu$. Can we say that there exists a finite valued measurable function $f$ on $\Omega$ such that: $\nu(E)=\int_Efd\mu$ (Like what we have in the Radon-Nikodym Theorem)?