Last Updated：2025/11/26
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When constructing the μ-completion of the measurable space, we adjoin all subsets that lie between two measurable sets of equal measure so that the measure becomes complete as well.
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When constructing the μ-completion of the measurable space, we adjoin all subsets that lie between two measurable sets of equal measure so that the measure becomes complete as well.
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Related words
μ-completion
Noun
(mathematical
analysis)
A
σ-algebra
which
is
obtained
as
a
"completion"
of
a
given
σ-algebra,
which
includes
all
subsets
of
the
given
measure
space
which
simultaneously
contain
a
member
of
the
given
σ-algebra
and
are
contained
by
a
member
of
the
given
σ-algebra,
as
long
as
the
contained
and
containing
measurable
sets
have
the
same
measure,
in
which
case
the
subset
in
question
is
assigned
a
measure
equal
to
the
common
measure
of
its
contained
and
containing
measurable
sets
(so
the
measure
is
also
being
completed,
in
parallel
with
the
σ-algebra).
Japanese Meaning
（数学解析において）与えられたσ‐加法族（測度空間の可測集合の族）を補完する操作によって得られるσ‐加法族。つまり、もともとのσ‐加法族に含まれないが、ある内包集合と包含集合が同じ測度を持つ場合にその共通の測度を割り当てることで、補足的に定義される集合を加えた結果得られる完全なσ‐加法族。 / 測度の拡張という観点から、既存のσ‐加法族が持つ測度定義を補完し、測度空間内の例外的（零集合的な）部分集合も測度の一貫性を保って含める拡張されたσ‐加法族を表す。
Related Words
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