BorelMeasuresAndFunctions

A Borel measure and a Borel function are distinct concepts in measure theory and topology, though they both involve Borel sets.

Borel Measure

• Definition: A Borel measure is a measure defined on the σ-algebra of Borel sets of a topological space. It assigns a non-negative real number or infinity to each Borel set in a way that is countably additive[1][3].
• Properties: Borel measures can be regular, meaning they are both inner and outer regular. They may also be locally finite, especially in locally compact spaces[3].
• Applications: Borel measures are used to quantify the "size" of sets in a topological space, and they form the foundation for integration and probability on these spaces.

Borel Function

• Definition: A Borel function is a function between two topological spaces such that the inverse image of any open set in the target space is a Borel set in the domain space[5][6].
• Properties: Every continuous function is a Borel function, but not every Borel function is continuous. The class of Borel functions includes all functions whose inverse images of open sets are Borel, making them more general than continuous functions[4][5].
• Applications: Borel functions are important in analysis and probability because they ensure that certain operations (like integration) can be performed on them. They are used to define measurable functions in contexts where the domain or range is equipped with a Borel σ-algebra.

Comparison

Aspect	Borel Measure	Borel Function
Definition	A measure on the σ-algebra of Borel sets	A function with inverse images of open sets being Borel
Focus	Quantifying size or volume of sets	Ensuring measurability of functions
Properties	Countably additive, possibly regular or locally finite	Includes continuous functions; inverse images are Borel
Applications	Integration, probability measures	Analysis, measurable functions

In summary, while both concepts involve Borel sets, a Borel measure deals with assigning measures to these sets, whereas a Borel function concerns the measurability of functions based on their inverse images being Borel sets.

Citations: [1] https://encyclopediaofmath.org/wiki/Borel_measure [2] https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch3.pdf [3] https://en.wikipedia.org/wiki/Borel_measure [4] https://www-users.cse.umn.edu/~garrett/m/real/notes_2016-17/real-notes-02.pdf [5] https://encyclopediaofmath.org/wiki/Borel_function [6] https://www.qedcat.com/measuretheory/04-MEASURABLE%20FUNCTIONS.pdf