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In the mathematical discipline of measure theory, the intensity of a measure is the average value the measure assigns to an interval of length one.
Let be a measure on the real numbers. Then the intensity of is defined as
if the limit exists and is independent of for all .
Look at the Lebesgue measure . Then for a fixed , it is
so
Therefore the Lebesgue measure has intensity one.
The set of all measures for which the intensity is well defined is a measurable subset of the set of all measures on . The mapping
defined by
is measurable.
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