Measurable function

Let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be measurable spaces. A function:

is called measurable if for every $G\in \mathcal{G}$, the pre-image belongs to $\mathcal{F}$:

Special Cases:

1. Real/complex-valued measurable functions:

   • If $Y=\mathbb{R}$ or $\mathbb{C}$, $\mathcal{G}$ is the Borel $\sigma$-algebra.
   • It suffices to check measurability on a generating set (e.g., open intervals $(a,b)$ or strips $\{z\mid a<\text{Re}(z)<b\}$).

2. Random variables:

   • See random variable