Suppose I am given a countable family $(\mu\_n)$ of finite Borel-measures on a compact interval $[0,T]$. Can I find a dominating measure $\mu$ (with $\mu\_n \ll \mu$ for all $n$), such that all Radon-Nikodym densities $\frac{d\mu\_n}{d\mu}$ are essentially bounded by a constant $K$ (independent of n)?



Dominating measure with bounded Radon-Nikodym density
No, take any finite measure $\mu$ and look at the family $n \mu$ for n integral.


