Talk:Borel regular measure

Second item in definition

Hi, reading the second item of the definition sounds a bit rare to me:

For every set A ⊆ Rn (which need not be μ-measurable) there exists a Borel set B ⊆ Rn such that A ⊆ B and μ(A) = μ(B).

If we start saying that A need not to be μ-measurable, how is that at the end we have μ(A)? — Preceding unsigned comment added by Jorgeecardona (talk • contribs) 21:45, 8 September 2016 (UTC)[reply]

Hi, it is correct to use A only set. μ(A) is defined because μ is an outer measure. See for instance 'Geometric measure theory' by H. Federer or 'Measure theory and fine properties of functions' by L. C. Evans and R. F. Gariepy. Quantrillo (talk) 14:35, 25 March 2020 (UTC)[reply]

more generality?

The article Riesz representation theorem links to this page, referring to a Borel regular measure on a locally compact Hausdorff space, while on this page a Borel regular measure is only defined for $\mathbb {R} ^{n}$ . Seems like there exists a more general definition. 82.82.87.235 (talk) 22:23, 9 July 2009 (UTC)[reply]

merging section into content (measure theory)

hi guys,

given that the generation of the regular borel measure is among the central objects of study in measure theory, and the fact that one can start with a primitive notion (set theory) to build a content, which then can generate such a measure, i was hoping we could start some discussion on tidying up some pages.

i dislike definitions of measures that start without a primitive notion, as it defeats the purpose of the simplicity that measure theory is supposed to exploit.

this specific page (borel regular measure) is so brief i suspect there won't be much argument if we can merge it into content (measure theory). however, i thought it was a useful example of a page that's not really helpful (no details on construction etc).

having read Halmos' (beautiful) construction of a borel measure from a content, which is then used to build the theory of Baire measures and linear functionals, i feel that we need to have a discussion on how we can combine a bunch of these pages.

i do not think the likes of Constantin Carathéodory thought there would be more than one way to define a measure (aside from things like a Gaussian measure, which is the analytic form resulting from measuring "true nature").

Carathéodory's extension theorem is a very intimidating proof, but its beauty lies in the fact it can extend the appropriate definition (such as the one given by Halmos) into a Lebesgue measure.

concepts relevant to this proof, such as set cover, enable us to exploit the Hasse principle (global-local principle) to procure such a measure

think of a unit sphere centred at each point in space; the natural overlap between adjacent points' spheres ("connected") provides fertile ground to apply Caratheodory's extension theorem. not saying it's published, but it has been done and known for quite some time.

being someone who considers himself a (seasoned, mind you) novice in this intimidating area of mathematical analysis, some of these definitions seem superfluous (inner regular, outer regular, etc).

i accept that there are few resources to help fix these issues, but i thought i'd throw out my own view of some of these pages.

i am not trying to disparage contributions from mathematicians like Johann Radon and the likes, who have all tried to further develop this challenging line of thinking initiated by Henri Lebesgue, Hermann Minkowski and his student Constantin Carathéodory; rather, i am just trying to point out the fact that measure theory was *always* focused on *measuring* in the real world.

i plan to start reading Caratheodory's exposition that propounds a measure-theoretic approach to thermodynamics after i click "save page", but i suspect this work will directly support my previous sentence (although he is just one person, he is Constantin Caratheodory...).

i have belaboured the point that any numerical analysis purporting to use measure theory must demonstrate their construction starting from a primitive notion (sets), which are the foundation that later results are built on (see halmos).

you guys may disagree and shut down my line of thinking, which is fine, but i thought i'd throw it out there as i've noticed a lot of fragmentation. thanks 174.3.155.181 (talk) 22:02, 10 July 2016 (UTC)[reply]

Relationship with full Borel measures

I dislike the last paragraph. The fact that the restriction to the Borel σ-algebra gives a full Borel measure is not specific to Borel regular measures. It works fine for Borel measures as well. What's more interesting is that we have a one-to-one correspondance between full Borel measures and Borel regular outer measures. — Preceding unsigned comment added by 2A01:E35:2FB5:F070:1C24:79AD:2514:941B (talk) 11:01, 2 April 2020 (UTC)[reply]