Show that a space is a probability space. Let $X : \Omega \rightarrow \mathbb{R}$ be a non-negative random variable defined on a probability space $(\Omega, F, \mathbb{P})$ with $E[|X|]<\infty $. For $A\in F$ define:
$$Q[A]=\frac{E[X \chi_A]}{E[X]}$$
where $\chi_A$ is the indicator(characteristic) function of A. Is $(\Omega, F, Q)$ a probability space?
What we need to show is that $Q$ is a probability measure, so we have to check the three properties: non-negativity, normalization and countable additivity. I already checked the first two but I have no idea how to prove the last part.
Could anyone help?

Note that if $A_i$ are disjoint and $A$ is their union, then $1_A = \sum 1_{A_i}$.
Since $X$ is nonnegative 
$$E[X1_A] = E[X \sum_{i=1}^\infty 1_{A_i}] 1_{A_i}] =E[\lim_{N\to\infty}X \sum_{i=1}^N 1_{A_i}]= \lim_{N\to\infty}\sum_{i=1}^N E[X 1_{A_i}] =\sum_{i=1}^\infty E[X 1_{A_i}]$$
where here we have applied the monotone convergence theorem since the partial sums are increasing.
