Question

If $f:X\to \mathbb{R}$ and $f^{-1}((r,\mathrm{\infty }])\in \mathcal{M}$ for each $r\in \mathbb{Q}$, then $f$ is measurable.

Short answer

Function $f$ is measurable because preimages of intervals $(r,\mathrm{\infty }]$ for $r\in \mathbb{Q}$ determine the measurability of preimages of Borel sets.

Step-by-step solution

Understand the Problem

We need to prove that function $f$ is measurable if for every real number $r$, the preimage $f^{-1}((r,\mathrm{\infty }])$ is a measurable set in $\mathcal{M}$. Measurability of $f$ means that preimages of Borel sets under $f$ belong to the sigma-algebra $\mathcal{M}$.

Establish Definition of Measurability

A function $f:X\to \mathbb{R}$ is $\mathcal{M}$-measurable if for every Borel set $B\subset \mathbb{R}$, the preimage set $f^{-1}(B)$ is in $\mathcal{M}$. It suffices to check preimages of intervals of form $(-\mathrm{\infty },b)$.

Approximate Open Intervals Using Rational Numbers

To exploit the given condition, note that any open interval $(-\mathrm{\infty },b)$ can be expressed as a union of half-open intervals $(-\mathrm{\infty },r)$ for rational numbers $r<b$. This uses the density of rational numbers in real numbers.

Show Preimage of Open Intervals is Measurable

Consider $f^{-1}((-\mathrm{\infty },b))=\bigcup _{r<b,r\in \mathbb{Q}}{f}^{-1}((r,\mathrm{\infty }]{)}^c$. Each set $f^{-1}((r,\mathrm{\infty }]{)}^c$ is measurable because it is derived from measurable $f^{-1}((r,\mathrm{\infty }])$. Thus, their union is measurable.

Conclude Measurability of f

Since the preimage of any $(-\mathrm{\infty },b)$ is measurable and the open sets generate the Borel sigma-algebra, $f$ is measurable because it satisfies the required criterion for the preimage of Borel sets being measurable.

Key concepts

Borel sets

Borel sets are a central concept in measure theory and are essential when discussing measurable functions. The Borel sigma-algebra is generated by open sets on the real line. Essentially, Borel sets include all possible sets you can create using open intervals, closed intervals, and countable operations like unions and intersections.
One of the key properties of Borel sets is their role in defining measurable functions. When we say that a function is measurable, we are typically referring to whether the preimages of Borel sets under this function belong to a particular sigma-algebra that is defined over the space. Understanding Borel sets helps us grasp the kinds of conditions and sets we are working with in problems involving measurability.

• Borel sets are generated from open intervals in the real line.
• They are closed under countable unions and intersections.
• They form the smallest sigma-algebra containing all open intervals in the real line.

Knowing these facts allows mathematicians to work with intricate properties and operations involving functions and their representations.

sigma-algebra

A sigma-algebra is another fundamental concept in measure theory, providing the mathematical structure that helps us describe collections of sets with nice closure properties. In formal terms, a sigma-algebra over a set X is a collection of subsets of X that is closed under various operations, such as complement and countable unions.
This structure is crucial when defining measurable functions, as it determines the 'measurable sets' with which we are concerned. Whenever a function's preimage of a Borel set falls within a specified sigma-algebra, the function is considered measurable with respect to that sigma-algebra. Sigma-algebras play a pivotal role in defining measures, integrating functions, and understanding probabilistic events in a rigorous way.

• A sigma-algebra must include the empty set.
• If a set is in a sigma-algebra, then its complement must also be in the sigma-algebra.
• Sigma-algebras are closed under countable unions, meaning if you take the union of any count of sets within the sigma-algebra, the result is also inside of it.

These properties imply that sigma-algebras are an ideal framework for handling the complexities involved in defining and analyzing measurable processes.

preimage

In the context of functions and their measurement, the preimage concept is incredibly useful. The preimage of a set B under a function f is essentially the collection of all elements in the function’s domain that map to elements in B. Mathematically, for a function $f:X\to Y$, the preimage of a subset $B\subseteq Y$ is $f^{-1}(B)=\{x\in X\mid f(x)\in B\}$.
Preimages help us translate properties of sets in the codomain back to the domain. This translation is crucial when working with measurable functions, where we often need to show that preimages of Borel sets are measurable. In the exercise above, if preimages of half-open intervals are measurable under a given sigma-algebra, you'd conclude that the function is measurable. Understanding preimages links the structure of the codomain back to the domain, allowing you to apply properties of sigma-algebras effectively.

• The preimage of a set is always a subset of the function's domain.
• Knowing the preimage allows us to backtrack or reconstruct inputs that lead to certain outputs.
• Preimage concepts are pivotal in analyzing how functions transfer one set structure into another.

Using preimages helps us understand the relationships between different sets and provides a framework for exploring the behavior of functions in the realm of measure theory.