# When family of DF's $\mathcal{P}$ fail to be dominated by a measure $\mu$

On the topic of minimal sufficient statistics, there is an important theorem which requires the family of probability distributions $$\mathcal{P}$$ is dominated by some measure $$\mu$$.

As I understand it, "dominated" in this context simply means every $$p_\theta \in \mathcal{P}$$ is absolutely continuous with respect to $$\mu$$.

My question is, when is such a condition not fulfilled? It seems like, given any family of probability distributions, you could construct a measure $$\mu$$ that dominates $$\mathcal{P}$$

I feel like I've fundamentally misunderstood this. Could someone explain it to me properly?

For dominance, it is important that the dominating measure $$\mu$$ be $$\sigma$$-finite$$\dagger$$.
If your family of probability distributions $$\mathcal{P}$$ is countable, then yes, it is always dominated. You can take $$\mu=\sum_i 2^{-i} P_i$$. For a non-dominated example, take counting measures on some uncountable set of points.
$$\dagger$$ The reason is that the use of domination is mostly related to the Radon-Nikodym theorem, which requires $$\sigma$$-finite measures: Wikipedia: Radon-Nikodym Theorem.