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Let $(\Omega,\mathcal{F})$ be a measurable space. The following are equivalent:

  1. $\ X:\Omega \to \mathbb{R} $ is a Borel function.

  2. $\{\omega\in\Omega:X(\omega)>a\}\in\mathcal{F}$ for all $a\in\mathbb{R}$.

  3. $\{\omega\in\Omega:X(\omega)< a\}\in\mathcal{F}$ for all $a\in\mathbb{R}$.

  4. $\{\omega\in\Omega:X(\omega) \in B\}\in\mathcal{F}$ for all open subsets $B\subset\mathbb{R}$.

  5. $\{\omega\in\Omega:X(\omega) \in B\}\in\mathcal{F}$ for all closed subsets $B\subset\mathbb{R}$.

How on earth would I prove this? I have no idea where to start. Any help would be very much appreciated. Thanks

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