= Solution
{c}
A <partition of a sample space> is a finite or countable family $(A_j)$ of pairwise disjoint <events> whose union is the whole <sample space>. If $X$ is <integrable>, the <law of total expectation> states
$$
\boxed{\mathbb E[X]=\sum_{j:\mathbb P(A_j)>0}\mathbb P(A_j)\,\mathbb E[X\mid A_j].}
$$
This follows by summing $X\mathbf1_{A_j}$ over the <partition of a sample space>; absolute integrability justifies the countable sum. For nonnegative $X$, the identity also holds with infinite expectations. Zero-probability parts contribute zero without requiring a conditional value on them.
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