Equivalent Martingale (/Risk Neutral) Measure Conditions

Question

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.

Answer 1

The concept of arbitrage must be independent of the probability measure. This means that only measures equivalent to the real-world measure, where you can actually measure arbitrage, are permissible.

Answer 2

Two measures are said to equivalent if they have the same null sets. For finance, this means that both measures agree on which events can happen and which events can't. So, all this definitiion really imposes is that the real world measure and the risk-neutral measure agree on what events can take place and which ones can't.

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.