The fundamental theorem of asset pricing states that the existence of an equivalent martingale measure is equivalent to the absence of arbitrage opportunities in financial markets. This theorem is foundational for pricing by arbitrage, allowing the replacement of a physical probability measure with an equivalent martingale measure, enabling the use of martingale theory for fair pricing of contingent claims. The theory of martingale representation helps characterize assets that can be replicated through trading. Changing measures from P to Q reflects risk aversion, adjusting probabilities to favor unfavorable events. This concept has historical roots, such as in insurance using mortality tables. The paper explores the precise meaning of "essentially" in the theorem, addressing the relationship between no-arbitrage and equivalent martingale measures. It also investigates which stochastic processes can become martingales under an equivalent measure. The main theorem characterizes the existence of equivalent martingale measures in terms of "no free lunch with vanishing risk," emphasizing the necessity of general stochastic integration theory for processes with jumps. The theorem contributes to both economics and mathematics, highlighting the importance of no-arbitrage conditions and the complexity of stochastic processes. The process S, which is R-valued, can be extended to d-dimensional processes without significant changes. The paper discusses the general idea of no-arbitrage and its variants, emphasizing that no trading strategy should yield non-negative payoffs.The fundamental theorem of asset pricing states that the existence of an equivalent martingale measure is equivalent to the absence of arbitrage opportunities in financial markets. This theorem is foundational for pricing by arbitrage, allowing the replacement of a physical probability measure with an equivalent martingale measure, enabling the use of martingale theory for fair pricing of contingent claims. The theory of martingale representation helps characterize assets that can be replicated through trading. Changing measures from P to Q reflects risk aversion, adjusting probabilities to favor unfavorable events. This concept has historical roots, such as in insurance using mortality tables. The paper explores the precise meaning of "essentially" in the theorem, addressing the relationship between no-arbitrage and equivalent martingale measures. It also investigates which stochastic processes can become martingales under an equivalent measure. The main theorem characterizes the existence of equivalent martingale measures in terms of "no free lunch with vanishing risk," emphasizing the necessity of general stochastic integration theory for processes with jumps. The theorem contributes to both economics and mathematics, highlighting the importance of no-arbitrage conditions and the complexity of stochastic processes. The process S, which is R-valued, can be extended to d-dimensional processes without significant changes. The paper discusses the general idea of no-arbitrage and its variants, emphasizing that no trading strategy should yield non-negative payoffs.