= Solution
{c}
The <function> $\phi_a(t)=t\wedge a$ is a <concave function>, and $X\wedge a$ is <integrable> since $|X\wedge a|\leq|X|+a$. The concave form of the <conditional Jensen inequality> gives
$$
\mathbb E[X\wedge a\mid\mathcal G]
\leq\bigl(\mathbb E[X\mid\mathcal G]\bigr)\wedge a
=Y\wedge a\quad\text{almost surely}.
$$
The <random variables> $X$ and $Y$ have the same <probability distribution>, so $\mathbb E[X\wedge a]=\mathbb E[Y\wedge a]$. Using the <law of total expectation>, the nonnegative difference between the right and left sides of the inequality has zero <expectation>. It must vanish <almost surely>. Therefore
$$
\boxed{\mathbb E[X\wedge a\mid\mathcal G]=Y\wedge a\quad\text{almost surely}.}
$$
The equality comes from the distributional assumption as well as the <conditional Jensen inequality>.
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