Let $ {(S_t)}_{t\in[0,+\infty[} $ be a semimartingale and ${(x_t)}_{t \in[0,+\infty[}$ an admissible strategy. We denote by $(x.S)_{+\infty}=\lim \int_{0}^{t} x_u dS_u$ if such limit exists, and by $K_0$ the subset of $L^0(\Omega,\mathcal{A}_{\infty},P)$ containing all such $(x . S)_{+\infty}$. Then we define:
$C_0=K_0-L^0_+(\Omega,\mathcal{A}_{\infty},P)$
$C=C_0\cap L^{\infty}_+(\Omega,\mathcal{A}_{\infty},P)$
The market model satisfies:
$\bullet$ the no-arbitrage condition (NA) if and only if $C \cap L^\infty(\Omega, \mathcal{A}_\infty, P) = \{0\}$, and
$\bullet$ the no-free-lunch-with-vanishing-risk condition (NFLVR) if and only if $\overline{C} \cap L^\infty(\Omega, \mathcal{A}_\infty, P) = \{0\}$.
There is no arbitrage if only if $C \cap L^{\infty}(\Omega,\mathcal{A}_{\infty},P)=\{0\}$
Question I am having a hard time making sense of this. If I remove from $K_0$ all a.s. positive functions, I mean $L^0_{+}$, then $C_0\cap L^{\infty}_{+}$ must be zero once $L^{\infty}\subset L^0$ and hence $L^{\infty}_+\subset L^0_+$. How is this supposed to be a condition for arbitrage? This is always zero by the definition of the spaces. Am I understanding this correctly?
If you are reading The Mathematics of Arbitrage by Delbaen and Schachermayer, which I suspect you are, then you should note that they use $\setminus$ and not $-$ for set difference. For them the set you care about is defined as:
$$C_0 = K_0 - L_+^\infty = \{f-g \, | \, f\in K_0, f\in L^\infty, g \geq 0\}$$
Thus $C_0$ should be understood as the cone in $L_+^\infty$ of contingent claims dominated by the payoff of some admissible strategy. In financial terms, $C_0$ is the set of contingent claims which can be "superreplicated" incurring zero cost. That is, we are able to come up with a market portfolio to pay at least $f\in C_0$ and, if necessary, we dispose of some cash.