Let $\displaystyle X \, : \, (\Omega, \mathcal{A}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ be a random variable. Let

$\displaystyle \mathcal{F} = \{A \, : \, A = X^{-1}(B), \mbox{ some } B \in \mathcal{B}(\mathbb{R})\} $.

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

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

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