= Solution
Construct the strictly positive <rolling one-period bond account> using successive one-period <zero-coupon bonds>:
$$
\beta_0=1,\qquad\beta_t=\frac{\beta_{t-1}}{P_{t-1}(t)}.
$$
At time $t-1$, invest the entire account value in the bond maturing at $t$. This gives a positive <self-financing portfolio> usable as a <numéraire>.
We use the finite-discrete-time <fundamental theorem of asset pricing>: in a frictionless market with finitely many adapted assets and trading dates, no arbitrage is equivalent to existence of an <equivalent martingale measure> for asset prices, including dividends, expressed in a positive traded <numéraire>. Maturing bond payoffs are reinvested in that numéraire. Let $\mathbb Q$ be such a measure and $L_t=\mathbb E[d\mathbb Q/d\mathbb P\mid\mathcal F_t]$ its positive density process. The <martingale> pricing relation for a unit <zero-coupon bond> is
$$
\frac{P_t(T)}{\beta_t}=\mathbb E_{\mathbb Q}\left[\frac1{\beta_T}\mid\mathcal F_t\right].
$$
Define $Z_t=L_t/\beta_t$. The <Bayes formula for conditional expectation> then gives
$$
\boxed{P_t(T)=\frac{\mathbb E(Z_T\mid\mathcal F_t)}{Z_t}.}
$$
The positive expectations are finite because these are the traded finite bond prices; in particular $\mathbb EZ_T=Z_0P_0(T)$ when initial information is trivial. Normalize $Z_0=1$ without changing any ratio. With nontrivial initial information the same identities are conditional on that information. The <state-price density> need not be unique when the bond market is incomplete; existence is sufficient.
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