Let's say we have 3 binary independent random variables $X_k$, e.g. each one represents if an event happens or not. he possible values of a $k$-th variable are $0$ and $m_k$. Then, if I am not mistaken, the state space is given by
$$ \Omega = \{123, 12\overline{3}\, 1\overline{2}3\, \overline{1}23, 1\overline{23}, \overline{1}2\overline{3}, \overline{12}3, \overline{123} \} $$
where $\overline{k}$ denotes event $k$ not happening.
Define $Y = X_1+X_2+X_3$. By the law of total expectation, if I am not mistaken,
$$ E[Y] = \sum E\left[Y|\omega_i\right]P(\omega_i) \\= (m_1+m_2+m_3)P(123) + (m_1+m_2)P(12\overline{3})+(m_1+m_3)P(1\overline{2}3) + m_1P(1\overline{23}) + ... \\= m_1 \left( P(123) + P(12\overline{3}) + P(1\overline{2}3) + P(1\overline{23}) \right) + ... \\= m_1 \left( P(12) + P(1\overline{2}) \right) + ... \\= m_1P(1) + m_2P(2) + m_3P(3) $$
where in the intermediate steps above I am only showing the terms involving $m_1$ for readability. The results is of course nothing more than a rule for the expectation of a sum of RVs, but how do I interpret it terms of the partition of the original sample space $\Omega$ and the law of total expectation?