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

The pushforward measure

Given a measure $\mu$ in $(X,\mathcal{F})$ we can define a new measure $\nu$ on $(Y,\mathcal{G})$ by setting:

for all $B\in \mathcal{G}$.

This $\nu$ is called the pushforward measure or image measure, often denoted $f_{\ast }\mu$ or $\mu \circ f^{-1}$.

This pushforward construction is essential in probability theory (where it describes how random variables induce probability distributions) and in many areas of analysis and geometry.