= Solution
For each terminal event $A$, <market completeness> supplies a portfolio $H_A$ replicating its <indicator function>. The two pricing identities give
$$
\mathbb E[X\mathbf1_A]=H_A^T\mathbb E[XP_1]=H_A^TP_0
=H_A^T\mathbb E[YP_1]=\mathbb E[Y\mathbf1_A].
$$
The constant payoff is also replicable, so these positive pricing variables are integrable under the stated finite pricing expectations. Equivalently, the preceding finite-atom result makes them finite-valued modulo null sets. Taking $A=\{X>Y\}$, the equality says that the nonnegative variable $(X-Y)\mathbf1_A$ has expectation zero; thus $X\le Y$ almost surely. Reversing the roles gives
$$
\boxed{X=Y\quad\text{almost surely}.}
$$
This is <uniqueness of a one-period pricing density> in a <complete market>. It uses replication of every event, not only matching a few marginal asset expectations.
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