The chapter introduces the fundamental theorem of asset pricing, a cornerstone in mathematical finance. This theorem states that the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage opportunities. The process $(S_t)_{t \in \mathbb{R}_+}$ represents the random evolution of discounted asset prices. The theorem allows for the replacement of the underlying probability measure $\mathbf{P}$ with an equivalent measure $\mathbf{Q}$, making $S$ a martingale under $\mathbf{Q}$. This enables the use of martingale theory to price contingent claims, a method known as the "equivalence principle" in actuarial science. The theory of martingale representation helps characterize assets that can be replicated through trading basic assets. The change from $\mathbf{P}$ to $\mathbf{Q}$ can also reflect risk aversion, giving more weight to unfavorable events. The paper explores the precise meaning of "essentially" in the theorem, aiming to understand the limitations and extensions of the approach. It discusses the economic significance of vanishing risk and the necessity of advanced stochastic integration theory for processes with jumps. The main theorem of the paper contributes to both mathematics and economics, characterizing the existence of an equivalent martingale measure for a general class of processes and emphasizing the need for sophisticated integration theory. The proof of the main theorem is complex, requiring advanced techniques from stochastic processes, functional analysis, and technical estimates. The concept of no-arbitrage and its weakenings are discussed, emphasizing the importance of trading strategies that do not yield non-negative payoffs.The chapter introduces the fundamental theorem of asset pricing, a cornerstone in mathematical finance. This theorem states that the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage opportunities. The process $(S_t)_{t \in \mathbb{R}_+}$ represents the random evolution of discounted asset prices. The theorem allows for the replacement of the underlying probability measure $\mathbf{P}$ with an equivalent measure $\mathbf{Q}$, making $S$ a martingale under $\mathbf{Q}$. This enables the use of martingale theory to price contingent claims, a method known as the "equivalence principle" in actuarial science. The theory of martingale representation helps characterize assets that can be replicated through trading basic assets. The change from $\mathbf{P}$ to $\mathbf{Q}$ can also reflect risk aversion, giving more weight to unfavorable events. The paper explores the precise meaning of "essentially" in the theorem, aiming to understand the limitations and extensions of the approach. It discusses the economic significance of vanishing risk and the necessity of advanced stochastic integration theory for processes with jumps. The main theorem of the paper contributes to both mathematics and economics, characterizing the existence of an equivalent martingale measure for a general class of processes and emphasizing the need for sophisticated integration theory. The proof of the main theorem is complex, requiring advanced techniques from stochastic processes, functional analysis, and technical estimates. The concept of no-arbitrage and its weakenings are discussed, emphasizing the importance of trading strategies that do not yield non-negative payoffs.