This seems like an obvious question but I can't seem to find any lemma online or in my probability textbook stating it. Basically I am wondering if the following statement is true:
$$ \sum_{a\in A} P(a|B) = 1 $$
For some event B, not independent of A.
Any insight or links to a proof would be great.
No, if $A$ is a discrete set then $\sum_{a \in A} P(\{a\} \mid B) = P(A \mid B)$, which is not necessarily $1$.
What is true is that $P(\cdot \mid B)$ is a probability measure, so in particular $\sum_{b \in B} P(\{b\} \mid B) = 1$.