Examples Regular measure

1 examples

1.1 regular measures
1.2 inner regular measures not outer regular
1.3 outer regular measures not inner regular
1.4 measures neither inner nor outer regular

examples
regular measures

lebesgue measure on real line regular measure: see regularity theorem lebesgue measure.
any baire probability measure on locally compact σ-compact hausdorff space regular measure.
any borel probability measure on locally compact hausdorff space countable base topology, or compact metric space, or radon space, regular.

inner regular measures not outer regular

an example of measure on real line usual topology not outer regular measure μ



μ
(
∅
)
=
0


{\displaystyle \mu (\emptyset )=0}

,



μ

(
{
1
}
)

=
0




{\displaystyle \mu \left(\{1\}\right)=0\,\,}

, ,



μ
(
a
)
=
∞




{\displaystyle \mu (a)=\infty \,\,}

other set



a


{\displaystyle a}

.
the borel measure on plane assigns borel set sum of (1-dimensional) measures of horizontal sections inner regular not outer regular, every non-empty open set has infinite measure. variation of example disjoint union of uncountable number of copies of real line lebesgue measure.
an example of borel measure μ on locally compact hausdorff space inner regular, σ-finite, , locally finite not outer regular given bourbaki (2004, exercise 5 of section 1) follows. topological space x has underlying set subset of real plane given y-axis of points (0,y) points (1/n,m/n) m,n positive integers. topology given follows. single points (1/n,m/n) open sets. base of neighborhoods of point (0,y) given wedges consisting of points in x of form (u,v) |v − y| ≤ |u| ≤ 1/n positive integer n. space x locally compact. measure μ given letting y-axis have measure 0 , letting point (1/n,m/n) have measure 1/n. measure inner regular , locally finite, not outer regular open set containing y-axis has measure infinity.

outer regular measures not inner regular

if μ inner regular measure in previous example, , m measure given m(s) = infu⊇s μ(u) inf taken on open sets containing borel set s, m outer regular locally finite borel measure on locally compact hausdorff space not inner regular in strng sense, though open sets inner regular inner regular in weak sense. measures m , μ coincide on open sets, compact sets, , sets on m has finite measure. y-axis has infinite m-measure though compact subsets of have measure 0.
a measurable cardinal discrete topology has borel probability measure such every compact subset has measure 0, measure outer regular not inner regular. existence of measurable cardinals cannot proved in zf set theory (as of 2013) thought consistent it.

measures neither inner nor outer regular

the space of ordinals @ equal first uncountable ordinal Ω, topology generated open intervals, compact hausdorff space. measure assigns measure 1 borel sets containing unbounded closed subset of countable ordinalls , assigns 0 other borel sets borel probability measure neither inner regular nor outer regular.