Conditional expectation as expectation $X$ is a real-valued random variable.
$B$ is an event.
Is it right that the conditional expectation $E[X\mid B]$ can be seen simply as an ordinary expectation over a new probability space having as a measure the corresponding conditional probability?
Thanks

What @Stefan Hansen said is enough. Just wanted to add that, if $X\geq 0$ or $\mathbb{E}[|X|1_B]<\infty$, then we can write
$$ \mathbb{E}[X|B] = \mathbb{E}[X\cdot P(B)^{-1}1_B],$$
where obviously $P(B)^{-1}$ can be taken out of expectation. This makes the conditional expectation (on an event) of a variable look just like expectation of a different variable, but under the same measure. Here, $1_B$ is $1$ if $B$ takes place and $0$ otherwise.
Best regards
