# Equivalent Martingale (/Risk Neutral) Measure Conditions

I am trying to understand EMM's and wanted to understand why we use EMM's over just martingale measures. The way we define EMM's is (for a simple one period model):

Given a probability measure $$\mathbb{P},$$

1. $$\mathbb{Q}$$ must be a probability measure,
2. $$\mathbb{E}^{\mathbb{Q}}[\bar{V_1}]=\bar{V_0}$$ where $$V_i$$ is the portfolio value at times $$0 \text{ and } 1$$,
3. $$\mathbb{Q}\sim \mathbb{P}$$ in the sense that they both have the same null sets.

I have seen the usefulness of assumption two and the first assumption seems like a core requirement to the framework, but I would like to know more about why the third assumption is useful. I have been told that it is useful when transfering statements about being $$\mathbb{P}$$ almost surely and $$\mathbb{Q}$$ almost surely.

For reference, we define a martingale measure as just the first two conditions.

In a simple discrete space setting, you typically assume $$\mathbb{P}$$ to be a strictly positive measure (well, unless $$\emptyset$$ obviously). i.e. $$\mathbb{P}[\{\omega\}]>0$$ for all $$\omega\in\Omega$$. So basically, you don't include any impossible states of world. This also means that statements hold for all states and not just $$\mathbb{P}$$-as. As you know, you can define the risk-neutral measure $$\mathbb{Q}$$ via Arrow-Debreu prices - and they should obviously positive as well. So, $$\mathbb{Q}$$ is a strictly posiitve measure as well and thus $$\mathbb{P}\sim\mathbb{Q}$$ is kind of natural.