Inmathematics, aregular measure on atopological space is ameasure for which everymeasurable set can be approximated from above by open measurable sets and from below by compact measurable sets.

Let (X, T) be a topological space and let Σ be aσ-algebra onX. Letμ be a measure on (X, Σ). A measurable subsetA ofX is said to beinner regular if

${\displaystyle \mu (A)=sup\{\mu (F)\mid F\subseteq A,F\text{ compact and measurable}\}}$

This property is sometimes referred to in words as "approximation from within by compact sets." Some authors^[1][2] use the termtight as asynonym for inner regular. This use of the term is closely related totightness of a family of measures, since afinite measureμ is inner regularif and only if, for allε > 0, there is somecompact subsetK ofX such thatμ(X \K) < ε. This is precisely the condition that thesingleton collection of measures {μ} is tight.

It is said to beouter regular if

${\displaystyle \mu (A)=inf\{\mu (G)\mid G\supseteq A,G\text{ open and measurable}\}}$

• A measure is calledinner regular if every measurable set is inner regular. Some authors use a different definition: a measure is called inner regular if everyopen measurable set is inner regular.
• A measure is called outer regular if every measurable set is outer regular.
• A measure is called regular if it is outer regular and inner regular.

• TheLebesgue measure on thereal line is a regular measure: see theregularity theorem for Lebesgue measure.
• AnyBaireprobability measure on anylocally compactσ-compact Hausdorff space is a regular measure.
• AnyBorel probability measure on a locally compact Hausdorff space with a countable base for its topology, or compact metric space, orRadon space, is regular.

• An example of a measure on the real line with its usual topology that is not outer regular is the measure${\displaystyle \mu }$ where${\displaystyle \mu (\mathrm{\varnothing })=0}$,${\displaystyle \mu \left(\{1\}\right)=0}$, and${\displaystyle \mu (A)=\mathrm{\infty }}$ for any other set${\displaystyle A}$.
• The Borel measure on the plane that assigns to any Borel set the sum of the (1-dimensional) measures of its horizontal sections is inner regular but not outer regular, as every non-empty open set has infinite measure. A variation of this example is a disjoint union of an uncountable number of copies of the real line with Lebesgue measure.
• An example of a Borel measure${\displaystyle \mu }$ on a locally compact Hausdorff space that is inner regular, σ-finite, and locally finite but not outer regular is given byBourbaki (2004, Chapter IV, Exercise 5 of section 1) as follows. The topological space${\displaystyle X}$ has as underlying set the subset of the real plane given by they-axis${\displaystyle \{0\}\times \mathbb{R}}$ together with the points (1/n,m/n²) withm,n positive integers. The topology is given as follows. The single points (1/n,m/n²) are all open sets. A base of neighborhoods of the point (0,y) is given by wedges consisting of all points inX of the form (u,v) with |v − y| ≤ |u| ≤ 1/n for a positive integern. This spaceX is locally compact. The measure μ is given by letting they-axis have measure 0 and letting the point (1/n,m/n²) have measure 1/n³. This measure is inner regular and locally finite, but is not outer regular as any open set containing they-axis has measure infinity.

• Ifμ is the inner regular measure in the previous example, andM is the measure given byM(S) = inf_U⊇S μ(U) where the inf is taken over all open sets containing the Borel setS, thenM is an outer regular locally finite Borel measure on a locally compact Hausdorff space that is not inner regular in the strong sense, though all open sets are inner regular so it is inner regular in the weak sense. The measuresM andμ coincide on all open sets, all compact sets, and all sets on whichM has finite measure. They-axis has infiniteM-measure though all compact subsets of it have measure 0.
• Ameasurable cardinal with the discrete topology has a Borel probability measure such that every compact subset has measure 0, so this measure is outer regular but not inner regular. The existence of measurable cardinals cannot be proved in ZF set theory but (as of 2013) is thought to be consistent with it.

• The space of all ordinals at most equal to the first uncountable ordinal Ω, with the topology generated by open intervals, is a compact Hausdorff space. The measure that assigns measure 1 to Borel sets containing an unbounded closed subset of the countable ordinals and assigns 0 to other Borel sets is a Borel probability measure that is neither inner regular nor outer regular.

• Borel regular measure
• Radon measure
• Regularity theorem for Lebesgue measure

• Billingsley, Patrick (1999).Convergence of Probability Measures. New York: John Wiley & Sons, Inc.ISBN 0-471-19745-9.
• Bourbaki, Nicolas (2004).Integration I. Springer-Verlag.ISBN 3-540-41129-1.
• Parthasarathy, K. R. (2005).Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. p. xii+276.ISBN 0-8218-3889-X.MR 2169627 (See chapter 2)
• Dudley, R. M. (1989).Real Analysis and Probability. Chapman & Hall.

• Measures (measure theory)

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