Glossary of economics research
Results of search for measure follow:
measure:
A noun, in the mathematical language of measure theory: a measure is a
function from sets to the real line. Probability is a common kind of measure
in economic models. Other measures are the counting measure, which is the
number of elements in the set, the length measure, the area measure, and the
volume measure. Length, area, and volume are defined along lines, planes, and
spaces just as one would expect, and they have the natural meanings.
Formally: a measure is a mapping m from a sigma algebra A to the
extended real line such that
(i) m(null) = 0
(ii) m(B) >= 0 for all B in A
(iii) m(any countable union of disjoint sets in A) = the sum of m(each
of those sets)
The third property is called the countable additivity property.
An example: imagine probability mass distributed evenly on a unit square.
Probability is then defined on any area within the square. The measure
(probability, here) is the size (area) of the subset.
The kinds of subsets on which measures such as probability are defined are
called sigma-algebras (which see).
Contexts: math; measure theory; real analysis
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