Before revision: The fundamental theorem of utility maximization (called FTUM hereafter) says that the utility maximization admits solution if and only if there exists an equivalent martingale measure.
After revision: This paper proposes two approaches that quantify the exact relationship among the viability, the absence of arbitrage, and/or the existence of the num\'eraire portfolio under minimal assumptions and for general continuous-time market models. Precisely, our first and principal contribution proves the equivalence among the utility maximization admits solution if and only if there exists an equivalent martingale measure.