Conditioning a random variable on an event
Not trying to be criticising.
I am cross-reading the same chapter from different books, and I felt those formulas were integral with respect to the set that is not consistent, or my understanding went wrong a bit, I am not sure are those considered correct?
For example, the first integration: I think that should be all Borel image of $B$ in $\mathbb{R}$ that is defined by $X$
Are those writing considered acceptable?
Sincerely ask for what you considered the correct form, respectively.
Here is what I think $$ P(B|A)=\int_{\hat{B}}f_{X|A}(x)dx\quad\text{for any borel}\,\hat{B}\subset\mathbb{R} \,\text{defined by} X(B)=\hat{B} $$