Let $\mathcal{F}$ be the set of all continuous probability densities.
I want a sub-family $\mathcal{F}_0$ of $\mathcal{F}$ that has a particular property (irrelavant here) such that every null-set of $\mathcal{F}_0$ is a null-set of $\mathcal{F}$.
It turns out that the location family of densities $Cauchy(\theta,1)$ has this particular property I'm after. All that remains is to verify the null-set condition.
Now, this family of Cauchy densities has support on the entire real line. So, is it correct to say this implies every null set of $\mathcal{F}_0$ is a null set of $\mathcal{F}$? Since the family of densities in $\mathcal{F}$ is restricted to having support only on the real line? So it's impossible to construct a null set of $\mathcal{F}_0$ that is not also a null set of $\mathcal{F}$?