Before revision:  As application of our version of the FTUM, we establish equivalence between the No-Unbounded-Profit-with-Bounded-Risk condition  , the existence of  num\'eraire portfolio, and the existence of solution to the utility maximization under equivalent probability measure  .
After revision:  As application of our version of the FTUM, we establish equivalence between the No-Unbounded-Profit-with-Bounded-Risk condition , the existence of num\'eraire portfolio, and the existence of the optimal portfolio under an equivalent probability measure .