Measurable Functions Determine Measurable Sets

Theorem
Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $f, g: X \to \overline{\R}$ be $\Sigma$-measurable functions.

Then the following sets are measurable:


 * $\left\{{f < g}\right\}$
 * $\left\{{f \le g}\right\}$
 * $\left\{{f = g}\right\}$
 * $\left\{{f \ne g}\right\}$

where, for example, $\left\{{f < g}\right\}$ is short for $\left\{{x \in X: f \left({x}\right) < g \left({x}\right)}\right\}$.