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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?

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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.

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