Risk-neutral pricing: the discounted asset price under the risk-neutral probability measure is a martingale, and it is the most important property of it. Besides, the uniqueness of risk neutral measure guarantees the market is complete, hence there’s no arbitrage. Otherwise, there’s no need to price the derivative if an arb exists.
To introduce Risk Neutral Prob, we used girsanov’s theorem to change the measure, the idea is introducing random variable Z for which the $E Z = 1$, and $\int {\mathbb{P}}(A) = \int_{A} Z(\omega) \overline{\mathbb{P}}(\omega), \text{for \\} \forall A \in \mathbb{F}$