Last Updated：2025/12/05
(mathematical analysis, measure theory) A measure-like function derived from some premeasure as follows: when it is applied to a set E, it yields the infimum of the premeasures of all possible premeasurable covers of E. More abstractly, a measure-like function whose domain is the power set of some underlying set, which has the properties of (1) yielding zero when applied to the empty set, (2) being monotonic (if A is subset of B then this function applied to A is less-than-or-equal to this function applied to B), (3) being subadditive (if A is a countable sum of Bⱼ's then this function applied to A is less-than-or-equal to the sum of the applications of this function to the Bⱼ's).
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outer measure
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Source Word
outer measure
Noun
(mathematical
analysis,
measure
theory)
A
measure-like
function
derived
from
some
premeasure
as
follows:
when
it
is
applied
to
a
set
E,
it
yields
the
infimum
of
the
premeasures
of
all
possible
premeasurable
covers
of
E.
More
abstractly,
a
measure-like
function
whose
domain
is
the
power
set
of
some
underlying
set,
which
has
the
properties
of
(1)
yielding
zero
when
applied
to
the
empty
set,
(2)
being
monotonic
(if
A
is
subset
of
B
then
this
function
applied
to
A
is
less-than-or-equal
to
this
function
applied
to
B),
(3)
being
subadditive
(if
A
is
a
countable
sum
of
Bⱼ's
then
this
function
applied
to
A
is
less-than-or-equal
to
the
sum
of
the
applications
of
this
function
to
the
Bⱼ's).
Japanese Meaning
外側測度: 測度論において、あるプリアィメジャーから派生して定義される関数で、ある集合に対して、その集合を覆うすべてのプリアィメジャ可能な被覆のプリアィメジャーの下限を取るもの。 / 性質として、空集合に対してゼロを与え、単調性（ある集合が別の集合の部分集合なら、その外側測度もそれに対応して小さいか等しくなる）や、可算部分加法性（無限個の集合の和集合に対して、その外側測度は各集合の外側測度の和以上にならない）を満たす。
Sense（1）
(mathematical
analysis,
measure
theory)
A
measure-like
function
derived
from
some
premeasure
as
follows:
when
it
is
applied
to
a
set
E,
it
yields
the
infimum
of
the
premeasures
of
all
possible
premeasurable
covers
of
E.
More
abstractly,
a
measure-like
function
whose
domain
is
the
power
set
of
some
underlying
set,
which
has
the
properties
of
(1)
yielding
zero
when
applied
to
the
empty
set,
(2)
being
monotonic
(if
A
is
subset
of
B
then
this
function
applied
to
A
is
less-than-or-equal
to
this
function
applied
to
B),
(3)
being
subadditive
(if
A
is
a
countable
sum
of
Bⱼ's
then
this
function
applied
to
A
is
less-than-or-equal
to
the
sum
of
the
applications
of
this
function
to
the
Bⱼ's).
（ plural ）
