# If there is a $T$-forward measure and a risk neutral measure, then markets are not complete?

I am trying to understand the connection between market completeness and risk neutral measures.

A market is complete if and only if the equivalent martingale measure is unique. But if I change to the $$T$$-forward measure ($$\mathbb{Q}^T$$), I have an equivalent measure to the money market account numéraire measure ($$\mathbb{Q}$$), so two different but equivalent measures. So the market is not complete?

In the case of an incomplete market, you no longer have a unique price for unattainable contingent claim, but rather a range of prices : $$\left(-p\left(-G\right), p\left(G\right)\right)$$ (see Bouchard & Chassagneux for notation) which correspond to the sub- and super-replication prices (lowest and highest prices beyond which the buyer or seller can make an arbitrage) for the European payoff $$G$$. If $$\mathcal{Q}$$ is the set of risk-neutral measures, $$-p \left(-G\right) = \inf\limits_{\mathbb{Q} \in \mathcal{Q}}{\mathbb{E}^\mathbb{Q} \left(e^{- \int_0^T{r_t \mathrm{d}t}}G\right)}$$ and $$p \left(G\right) = \sup\limits_{\mathbb{Q} \in \mathcal{Q}}{\mathbb{E}^\mathbb{Q} \left(e^{- \int_0^T{r_t \mathrm{d}t}}G\right)}$$ : there, you can see that there are infinitely many risk-neutral measures, and to each one corresponds a price within the price range.
The $$T$$-forward measure is not another risk-neutral measure: the asset used as numeraire is different (the $$T$$-zero coupon bond vs. the money market account). For any risk-neutral measure $$\mathbb{Q} \in \mathcal{Q}$$, you can get an equivalent $$T$$-forward measure $$\mathbb{Q}^T \in \mathcal{Q}^T$$, and you have similarly that $$-p \left(-G\right) = B\left(0, T\right)\inf\limits_{\mathbb{Q}^T \in \mathcal{Q}^T}{\mathbb{E}^{\mathbb{Q}^T} \left(G\right)}$$ and $$p \left(G\right) = B\left(0, T\right)\sup\limits_{\mathbb{Q}^T \in \mathcal{Q}^T}{\mathbb{E}^{\mathbb{Q}^T} \left(G\right)}$$.
If the market is complete, the set of risk-neutral measures is $$\mathcal{Q} = \left\{\mathbb{Q}\right\}$$, the set of $$T$$-forward measures is $$\mathcal{Q}^T = \left\{\mathbb{Q}^T\right\}$$ and the price of a European contingent claim with payoff $$G$$ is given by $$V_0 = \mathbb{E}^\mathbb{Q} \left(e^{- \int_0^T{r_t \mathrm{d}t}}G\right) = B \left(0, T\right)\mathbb{E}^{\mathbb{Q}^T} \left(G\right)$$