Peruse literature available and you’d probably get a bit confused with the differences between the First Fundamental Theorem and the Second Fundamental Theorem of Asset Pricing, and, the Completeness Theorem. So on purpose I will state and reference those here in that exact sequence.
- First Fundamental Theorem. This has to do with the mere EXISTENCE of a risk measure $\mathbb{Q}$ for primitive assets with no reference to derivatives.
- Second Fundamental Theorem. This highlights the importance of the UNIQUENESS of the martingale measure $\mathbb{Q}$.
- The Completeness Theorem. This ties together the two fundamental theorems with a key consequential result, namely that discounted asset prices are martingales under the exact same conditions that are necessary for the two theorems above to be fulfilled.
Now that we know what to expect we can state/quote the theorems as listed in various sources.