We denote discount factor
$D(t)$, and zero coupon bond
$B(t,T)$ as:
$$B(t,T) =\dfrac{1}{D(t)} E_t[D(T)]$$
here $E_t[X] = E[X|\mathcal{F}(t)].$
And we define
Zero curve
$Z(t,T)$
$$B(t, T)\cdot e^{Z(t,T)(T-t)} = 1$$
Libor
$L(t,T)$
$$B(t, T)\cdot (1 + (T-t) L(t, T)) = 1.$$
Denote $E^{T}[\ ]$ the $T$-forward measure i.e use $B(t,T)$ as numeraire.
I remember that Libor and zero curve should be martingale
under the $T$-forward measure. But we see the representations, equivalently $\dfrac{1}{B(t,T)}$ should be martingale under the $T$-forward. This is the $T$-forward price of $1,$ but discounted value of $1$ is not martingale under original measure i.e
$$E_t[D(T)\cdot 1] \neq D(t)\cdot1.$$
We can see that forward rate
$$B(t,T-\delta) = (1 + (T-t)F(t,T-\delta,T))B(t,T)$$
is really $T$-forward martingale, since $\frac{B(t,T-\delta)}{B(t,T)}$ is $T$-forward martingale, equivalently $D(t)B(t,T-\delta)$ is martingale under the original measure.
So, I really confuse here. Can anyone tell where is the mistake?