No free lunch with vanishing risk

For a semimartingale S, let ${\displaystyle K=\{(H\cdot S)_{\infty }:H{\text{ admissible}},(H\cdot S)_{\infty }=\lim _{t\to \infty }(H\cdot S)_{t}{\text{ exists a.s.}}\}}$ where a strategy is admissible if it is permitted by the market. Then define ${\displaystyle C=\{g\in L^{\infty }(P):\exists f\in K,~g\leq f~a.s.\}}$. S is said to satisfy no free lunch with vanishing risk if ${\displaystyle {\bar {C}}\cap L_{+}^{\infty }(P)=\{0\}}$ such that ${\displaystyle {\bar {C}}}$ is the closure of C in the norm topology of ${\displaystyle L_{+}^{\infty }(P)}$.[1]

If ${\displaystyle S=(S_{t})_{t=0}^{T}}$ is a semimartingale with values in ${\displaystyle \mathbb {R} ^{d}}$ then S does not allow for a free lunch with vanishing risk if and only if there exists an equivalent martingale measure ${\displaystyle \mathbb {Q} }$ such that S is a sigma-martingale under ${\displaystyle \mathbb {Q} }$.[2]