PreImage

The pre-image of a Borel set under a function is a fundamental concept in measure theory, a branch of mathematical analysis.

Borel Set

First, let's define what a Borel set is. In a topological space, a Borel set is any set that can be formed from open sets (or, equivalently, closed sets) through the operations of countable union, countable intersection, and relative complement. In the context of the real numbers, which are equipped with the standard topology, Borel sets include all open intervals, closed intervals, finite sets, countable sets, and many other complicated sets.

Pre-Image

Given a function $f: X \to Y$ and a set $B$ in $Y$, the pre-image (or inverse image) of $B$ under $f$ is the set of all elements in $X$ that $f$ maps into $B$. Formally, the pre-image of $B$ is defined as:

$$ f^{-1}(B) = { x \in X \mid f(x) \in B } $$

Pre-Image of a Borel Set

When we talk about the pre-image of a Borel set in the context of a measurable function, we're considering a function $f: X \to Y$ where $Y$ is typically the real numbers $\mathbb{R}$ (or a subset of it) endowed with its Borel σ-algebra (the collection of Borel sets). The set $f^{-1}(B)$ is the pre-image of a Borel set $B \subset Y$ under $f$.

A function $f$ is said to be measurable if the pre-image of every Borel set in $Y$ is a measurable set in $X$. In other words, for every Borel set $B$ in $Y$, the set of all $x$ in $X$ that map into $B$ under $f$ must be a measurable set according to the measure defined on $X$.

This concept is crucial in measure theory and is used to extend the notion of integrability to more complex functions than those in elementary calculus.