Measurability is preserved by sums, products, $\max$, $\min$, countable $\sup$, $\inf$, $\limsup$, $\liminf$, and pointwise limits.Suggest a slogan
Let \((X,\mcm)\) be a measurable space [@Fol13].
If \(f,g\colon X\to\RR\) are measurable, then \(f+g\), \(fg\), \(\max(f,g)\), and \(\min(f,g)\) are measurable.
If \(f_n\colon X\to[-\infty,\infty]\) are measurable for \(n\geq1\), then \(\sup_n f_n\), \(\inf_n f_n\), \(\limsup_n f_n\), and \(\liminf_n f_n\) are measurable; if \(f(x)\coloneqq\lim_n f_n(x)\) exists for every \(x\in X\), then \(f\) is measurable.
For \(E\subseteq X\), the characteristic function \(\chi_E\) is measurable if and only if \(E\in\mcm\).