I am reading some introductory notes to probability theory and I am puzzled by the sentence that follows. After having defined conditional probability as $$\text {Pr} (A|B) = \text {Pr}(A\cap B)/\text {Pr}B, $$ the author makes the following remark:
Another important aspect of the definition is that it maintains consistency between the original probability space and this new conditional space in the sense that for any events $A_1$, $A_2$ and any scalars $\alpha_1$ , $\alpha_2$, we have:$$\text {Pr}(\alpha_1 A_1 + \alpha_2 A_2 | B)= \alpha_1\text{Pr}(A_1|B)+\alpha _2 \text{Pr}(A_2|B)$$ This means that we can easily move back and forth between unconditional and conditional probability spaces.
I have totally no idea of what does this mean. Also, the author uses sometimes the notation $$A-B:=A\cap B^c$$ but has never defined something like $+$ or multiplication by scalars. What is this?
The source is this preliminary draft of a forthcoming book, page 10.
$A-B$ is the set-theoretic difference between $A$ and $B$ also often denoted by $A\setminus B$. Usually, it is defined by $$ A-B=\{x\in A\mid x\notin B\} $$ but this agrees with your definition since $$A\cap B^c=\{x\in A\mid x\in B^c\}=\{x\in A\mid x\notin B\}. $$
As to the equation $$ P(\alpha_1 A_1 + \alpha_2 A_2 \mid B)= \alpha_1P(A_1\mid B)+\alpha _2 P(A_2\mid B) $$ I can't make much sense of it when $A_1$ and $A_2$ are sets, since the right-hand side need not even be a probability (i.e. it can be greater than $1$ contradicting the equality). It does however make sense if you talk about the conditional expectation of random variables, but maybe that is out of the scope here.