What does it mean for a set to be Borel measurable?

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Any measure defined on the Borel sets is called a Borel measure.

How do you show a function is Borel measurable?

A function is measurable if the preimage of a measurable set is measurable. So that the preimages of Borel sets are Borel sets. If a function is continuous then it is also Borel measurable.

What does it mean for a function to be Lebesgue measurable?

Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated. In the case is Lebesgue measurable iff. is measurable for all. This is also equivalent to any of being measurable for all or the preimage of any open set being measurable.

Is continuous function measurable?

with Lebesgue measure, or more generally any Borel measure, then all continuous functions are measurable. In fact, practically any function that can be described is measurable.

Is a Borel measurable function Lebesgue measurable?

All Borel real valued functions on the euclidean space are Lebesgue-measurable, but the converse is false.

Is Lebesgue measure Borel regular?

The Lebesgue outer measure on Rn is an example of a Borel regular measure. It can be proved that a Borel regular measure, although introduced here as an outer measure (only countably subadditive), becomes a full measure (countably additive) if restricted to the Borel sets.

Are all Borel sets measurable?

The collection of Borel sets is the smallest sigma-algebra which contains all of the open sets. Every Borel set, in particular every open and closed set, is measurable. Therefore the collection of all measurable sets is a sigma-algebra.

Are all continuous functions Borel measurable?

In particular, every continuous function between topological spaces that are equipped with their Borel σ-algebras is measurable.

Is every Borel measure regular?

Let X be a metric space. Then every Borel measure μ on X is regular (i.e. for every Borel set B and every ε > 0, there exists a closed set Fε such that Fε⊂B and μ(B\ Fε) < ε).

What is the Borel measurability of a function?

The Borel measurability of the function f: X → Y is then equivalent to the measurability of the map f seen as map between the measurable spaces ( X, B ( X)) and ( Y, B ( Y)), see also Measurable mapping .

What is a Borel map?

A map f: X → Y between two topological spaces is called Borel (or Borel measurable) if f − 1 (A) is a Borel set for any open set A (recall that the σ -algebra of Borel sets of X is the smallest σ -algebra containing the open sets).

What is the difference between Baire and Borel functions?

In a general topological space the class of Baire functions might be strictly smaller then the class of Borel functions. Borel real-valued functions of one real variable can be classified by the order of the Borel sets; the classes thus obtained are identical with the Baire classes.

Is G\\Circ F a Borel function?

Moreover the compositions of Borel functions of one real variable are Borel functions. Indeed, if X, Y and Z are topological spaces and f:X o Y, g:Y o Z Borel functions, then g\\circ f is a Borel function, as it follows trivially from the definition above.