Is there a measure for every Borel σ-algebra?

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In summary, the Borel σ-algebra can be defined on every topological space and this allows every topological space to be turned into a measure space. However, this can be done in a trivial way by defining the measure to be 0 for all sets, or by using the counting measure or a point measure. A more comprehensive approach is through the Riesz representation theorem, which states that regular measures on a locally compact Hausdorff space can be represented by positive functionals. This is possible due to the Hahn-Banach theorem. The same can be achieved with C_0(X), where measures represent positive functionals and probability measures represent states, while point measures represent pure states. However, the existence of non
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Fredrik
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The Borel σ-algebra can be defined on every topological space. Does that mean that every topological space can be turned into a measure space?
 
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Yes, in a trivial way. Just define [itex]\mu(S)=0[/itex] for all S. Or take the counting measure. Or a point measure.

A better answer is actually Riesz representation theorem. This says that all regular measures on a locally compact Hausdorff space coincide with positive functionals

[tex]T:C_c(X)\rightarrow \mathbb{R}[/tex]

(with [itex]C_c(X)[/itex] the functions to [itex]\mathbb{R}[/itex] with compact support). Such a functional exist because of the Hahn-Banach theorem.

The same thing can be done with [itex]C_0(X)[/itex] actually. In that case: the measures are the positive functionals, the probability measures are the states and the point measures represent the pure states on X.

In general, the existence of non-trivial measures is not a simple problem and requires set theory (see for example measurable cardinals)
 
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  • #3
I wonder if the space has a cardinality greater than the continuum whether there are any measures other than the trivial one.
 
  • #4
lavinia said:
I wonder if the space has a cardinality greater than the continuum whether there are any measures other than the trivial one.

The first thing to do is to define the word "the trivial one".

This is quite an interesting theory. In general, this is studied in Boolean algebras and the so-called measure algebras.
 
  • #5
Excellent answer as always. Thanks. I've been meaning to study the Riesz representation theorem. It will be the first thing I do when I'm done with the basics of integration theory.
 
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1. What is a Borel σ-algebra?

A Borel σ-algebra is a type of mathematical structure used in measure theory to describe the collection of all possible measurable sets for a given topological space. It is generated by the open sets in the space and has important applications in probability and analysis.

2. Why is it important to have a measure for every Borel σ-algebra?

Having a measure for every Borel σ-algebra allows for a consistent and effective way to assign a numerical value to a set in a topological space. This is useful in applications such as determining probabilities or calculating integrals.

3. How is a measure defined for a Borel σ-algebra?

A measure for a Borel σ-algebra is typically defined as a function that assigns a non-negative real number to each set in the algebra, satisfying certain properties such as countable additivity and monotonicity. This allows for a consistent way to determine the "size" of a set.

4. Can a measure be defined for any Borel σ-algebra?

No, there are certain Borel σ-algebras for which a measure cannot be defined. These include non-measurable sets, which do not have a well-defined size, and pathological spaces, which do not have a consistent structure for defining a measure.

5. How is a measure for a Borel σ-algebra related to the concept of outer measure?

Outer measure is a more general concept that can be defined for any set, while a measure for a Borel σ-algebra is a specific type of outer measure that satisfies additional properties. In some cases, a Borel σ-algebra may have multiple measures, but there is only one outer measure for a given set.

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