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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?
lavinia said:I wonder if the space has a cardinality greater than the continuum whether there are any measures other than the trivial one.
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.
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.
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.
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.
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.