Lebesgue and Borel measurable functions

Let \(\mathcal{M}_L\) and \(\mathcal{M}_B\) be the \(\sigma\)-algebras of Lebesgue measurable and Borel subsets of \(\RR\), let \(E\in\mathcal{M}_L\), and let \(f\colon E \to [-\infty,\infty]\). The function \(f\) is Lebesgue measurable if \[ \theset{x \in E \suchthat f(x)>a}=f^{-1}((a, \infty]) \in \mathcal{M}_L \quad\text{for every } a\in\RR. \] When \(E\in\mathcal{M}_B\), the function \(f\) is Borel measurable if \(f^{-1}((a, \infty]) \in \mathcal{M}_B\) for every \(a\in\RR\).