# condition of risk neutral pricing

The theorem says if $$U$$ is a numeraire and let $$\mathbb{Q}^U$$ be the corresponding measure. Then for every tradable asset $$S$$, the relative price $$S_t/U_t$$ is a martingale under $$\mathbb{Q}^U$$. But I don't know the meaning of "tradable asset". For example: in Brigo's "interest rate models" p39: In this proof, $$\mathbb{E}^T$$ means the expectation under the forward measure( the numeraire is $$T$$-bond $$P(t,T)$$). $$\mathbb{E}$$ means the expectation under the risk-neutral measure (numeraire $$B(t)=e^{\int_0^tr_s\,ds}$$) I think he uses the risk-neutral pricing formula in the first equation: $$\mathbb{E}^T[r_T|\mathcal{F}_t]=r_t/P(t,T)$$ $$\mathbb{E}[e^{-\int_0^Tr_s\,ds}r_T|\mathcal{F}_t]=r_te^{-\int_0^tr_s\,ds}$$

my question is: 1. why the short rate $$r_t$$ is a tradable asset?

Let $$H$$ be the payoff function. In practice, I often apply the pricing formula as long as $$H\in L^2$$ regardless of whether it is a tradable asset or not. I know there is a complete market hypothesis: every derivative in the market is replicable.

my question is: 2 Does complete market implies $$\forall H\in L^2$$ is a tradable asset?

3 if the market is not complete, can we apply the pricing formula for $$r_t$$? i.e. how to judge $$H$$ is replicable or not?