Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $A\in \mathcal F$ s.t. $\mathbb P(A)>0$. In an exercise I have, I need to prove that $(A, \mathcal F_A, \mathbb P(.|A))$ is a probability space where $\mathbb P(C\mid A)=\frac{\mathbb P(A\cap C)}{\mathbb P(A)}$ and $\mathcal F_A=\{B\cap A\mid B\in \mathcal F\}$. Is it really true ?
I agree that $\mathcal F_A$ is a $\sigma -$agebra on $A$.
I agree that $\mathbb Q(C)=\frac{\mathbb P(C)}{\mathbb P(A)}$ for $C\in \mathcal F_A$ is a measure.
But for me $\mathbb P(.|A)$ is a measure on $\Omega $, and $(\Omega ,\mathcal F,\mathbb P(.|A))$ is a probability measure, but $(A,\mathcal F_A,\mathbb P(.|A))$ is not. So, what is $\mathbb P(.|A)$ for $(A,\mathcal F_A)$ ? or what represent the probability space $(\Omega ,\mathcal F,\mathbb P(.|A))$ for $(A,\mathcal F_A, \mathbb Q)$ ?