stochastic processes - About the definition of a complete market

In Steven Shreve's book "Stochastic Calculus for Finance 2", Definition 5.4.8 says a market is complete if every derivative security can be hedged. What exactly does every derivative security mean? The book up to Ch.5 has only considered European options. But in the proof of Theorem 5.4.9, it constructs a derivative security whose payoff $V(T)$ is path dependent. It looks like this is a case for American option. Is it that every payoff $V(T)$ which is a measurable function in $\mathcal{F}(T)$ is a derivative security, and every derivative security can be defined in such a way?