Measurability question

**TheProphet** · August 17th 2011, 05:13 AM

Let $X \, : \, (\Omega, \mathcal{A}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ be a random variable. Let
$\mathcal{F} = \{A \, : \, A = X^{-1}(B), \mbox{ some } B \in \mathcal{B}(\mathbb{R})\}$ .

Show that $X$ is measurable as a function from $(\Omega, \mathcal{F}) \mbox{ to } (\mathbb{R},\mathcal{B}(\mathbb{R}))$ .

Here $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$ -algebra on $\mathbb{R}$ .

Is it sufficient just to say that for every $B \in \mathcal{B}(\mathbb{R}), \, A = X^{-1}(B) \in \mathcal{F}$ , so $X$ is measurable from $(\Omega, \mathcal{F}) \mbox{ to } (\mathbb{R},\mathcal{B}(\mathbb{R}))$ by definition?