Radon Measure - Examples

Examples

The following are all examples of Radon measures:

• Lebesgue measure on Euclidean space (restricted to the Borel subsets);
• Haar measure on any locally compact topological group;
• Dirac measure on any topological space;
• Gaussian measure on Euclidean space with its Borel topology and sigma algebra;
• Probability measures on the σ-algebra of Borel sets of any Polish space. This example not only generalizes the previous example, but includes many measures on non-locally compact spaces, such as Wiener measure on the space of real-valued continuous functions on the interval .

The following are not examples of Radon measures:

• Counting measure on Euclidean space is an example of a measure that is not a Radon measure, since it is not locally finite.
• The space of ordinals at most equal to the first uncountable ordinal with the order topology is a compact topological space. The measure which equals 1 on any set that contains an uncountable closed set, and 0 otherwise, is Borel but not Radon.
• Let X be the interval [0,1) equipped with the topology generated by the collection of half open intervals . This topology is sometimes called Sorgenfrey line. On this topological space, standard Lebesgue measure is not Radon since it is not inner regular, since compact sets are at most countable.
• Let Z be the Bernstein set in (or any Polish space). Then no measure which vanishes at points on Z is a Radon measure, since any compact set in Z is countable.
• Standard product measure on for uncountable is not a Radon measure, since any compact set is contained within a product of uncountably many closed intervals, each of which is shorter than 1.