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.

Intuition for measurable functions

The formal definition of measurability can be understood through the intuition of information.
Think of the sigma-algebra $\mathcal{F}$ on the input space $X$ as the information we are allowed to have about the experiment's outcome, $\omega \in X$.

• You can imagine $\mathcal{F}$ as a "system of peepholes" into a dark room where the outcome $\omega$ is.
• This "holes system" $\mathcal{F}$ partitions the space $X$ into "chunks" or "atoms" (the smallest, indivisible sets in $\mathcal{F}$).
• When an outcome $\omega$ occurs, we cannot see $\omega$ itself, but we can see a light or something like that, indicating which "chunk" $\omega$ is in.

A function $f:X\to Y$ (like a random variable) is a calculation we want to perform based on the outcome $\omega$.

The "Constant on the Chunks" Test

The core intuition is this:

A function $f$ is $\mathcal{F}$-measurable if and only if it is constant on each of the information "chunks" defined by $\mathcal{F}$.

That is, once the experiment has happened, we can determine the value of $f$ with our holes system $\mathcal{F}$, since $f$ assigns the same value to all the results $\omega$ in the same chunk. So we don't worry about what the actual $\omega$ is, but to what subset in $\mathcal{F}$ it belongs to.

In other words, if we can't distinguish between two outcomes ${\omega }_1$ and ${\omega }_2$ (because they are in the same "chunk"), then an $\mathcal{F}$-measurable function $f$ cannot "illegally" distinguish them either—it must give them the same value, $f({\omega }_1)=f({\omega }_2)$.

Connecting Intuition to the Formal Definition

The formal definition is: $f^{-1}(G)\in \mathcal{F}$ for all $G\in \mathcal{G}$.

• Think of $G\in \mathcal{G}$ as a "question we can ask" about the output value $f(\omega )$. (e.g., "Is the output value in the interval (a, b)?")
• Think of $f^{-1}(G)$ as the set of all inputs $\omega$ that make the answer to that question "YES".

So, the definition $f^{-1}(G)\in \mathcal{F}$ means:

"For any question we can ask about the output (a set $G$), the set of all inputs that give a 'YES' answer (the set $f^{-1}(G)$) must be an event we can observe with our input information ($\mathcal{F}$)."

This is a direct result of the "constant on the chunks" rule. If $f$ is constant on the "chunks," any set $f^{-1}(G)$ will be a perfect union of these chunks, which is by definition an observable event in $\mathcal{F}$.