Measure

The terms "measure," "measurable," etc. have very precise technical definitions (usually involving sigma-algebras) that can make them appear difficult to understand. However, the technical nature of the definitions is extremely important, since it gives a firm footing to concepts that are the basis for much of analysis (including some of the slippery underpinnings of calculus).

For example, every definition of an integral is based on a particular measure: the Riemann integral is based on Jordan measure, and the Lebesgue integral is based on Lebesgue measure. The study of measures and their application to integration is known as measure theory.

A measure is defined as a nonnegative real function from a delta-ring 👁 F
such that

where 👁 emptyset
is the empty set, and

for any finite or countable collection of pairwise disjoint sets 👁 (A_n)
in 👁 F
such that 👁 A= union A_n
is also in 👁 F
.

If 👁 F
is 👁 sigma
-finite and 👁 m
is bounded, then 👁 m
can be extended uniquely to a measure defined on the 👁 sigma
-algebra generated by 👁 F
.

👁 m
is said to be a probability measure on a set 👁 X
if 👁 m(X)=1
and 👁 F
is a 👁 sigma
-algebra.

In the usual definition of a probability measure (or, more precisely a nontrivial 👁 sigma
-additive measure), a measure is a real-valued function 👁 mu
on the Power Set 👁 P(S)
of an infinite set 👁 S
that satisfies the following properties:

1. 👁 mu(emptyset)=0
and 👁 mu(S)=1
,

2. If 👁 X subset= Y
then 👁 mu(x)<=mu(Y)
,

3. 👁 mu({a})=0
for all 👁 a in S
(nontriviality),

4. If 👁 X_n,n=0,1,2,...
, are pairwise disjoint, then

(Jech 1997).

A measure 👁 m
may be extended by completion. The subsets of sets with measure zero form a 👁 delta
-ring 👁 G
. By "changing" sets in 👁 F
on a set from 👁 G
, a 👁 delta
-ring 👁 F_c
which is the completion of 👁 F
with respect to 👁 m
is obtained.

The measure 👁 m
is called complete if 👁 F=F_c
. If 👁 m
is not complete, it may be extended to 👁 F_c
by setting 👁 m((A\B) union (B\A))=m(A)
, where 👁 A in F
and 👁 B in G
.

See also

Almost Everywhere, Borel Measure, Ergodic Measure, Euler Measure, Gauss Measure, Haar Measure, Hausdorff Measure, Helson-Szegö Measure, Integral, Jordan Measure, Lebesgue Measure, Liouville Measure, Mahler Measure, Measurable Space, Measure Algebra, Measure Space, Minkowski Measure, Natural Measure, Probability Measure, Radon Measure, Wiener Measure Explore this topic in the MathWorld classroom

Portions of this entry contributed by Allan Cortzen

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References

Czyz, J. Paradoxes of Measures and Dimensions Originating in Felix Hausdorff's Ideas. Singapore: World Scientific, 1994.Jech, T. J. Set Theory, 2nd ed. Berlin: Springer-Verlag, p. 295, 1997.

Referenced on Wolfram|Alpha

Measure

Cite this as:

Cortzen, Allan and Weisstein, Eric W. "Measure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Measure.html

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