Talk:Measure (mathematics)

It is really enjoyable for a non mathematiciation to see this here, and easy to read too David Tribe 16:27, 25 January 2007 (CST)

remarks

Just a few thoughts to remember (how to reorganize this)

• separate particular examples from general classes (now Dirac measure is at the same logical level as Borel or Radon measure)
• sigma-finite and completeness are more or less at the same logical level (classes of measures)
• counterexamples should be moved to the lead to give some motivation or explanation for the need of the sigma-algebras.
• application - it would be nice to mention that some basic probability theory may be viewed as a direct application of the measure theory (identifying basic correspondence, definition of probability, types of convergence etc)

Aleksander Halicz 03:38, 7 February 2007 (CST)

You write

This does for instance happen in the real line case, where one expects any "natural" measure to be translation invariant. For such a measure there exists a set, which, if measurable, permits a direct proof of self-contradictory consequences, such as finite upper bound for an infinite sum of positive elements.

It might be that I don't follow your presentation, but are you thinking of a specific example. Of itself, such an infinite sum shouldn't pose a problem (i.e. sum of 1/2^n), so I think I'm missing something.

On a related note, I wonder how much we should include in the introductory section before a section of motivation is warranted. Still, I'm not certain if that's the most important thing right now.

Simen Rustad 15:52, 8 February 2007 (CST)

Thanks for your remakrs. Actually, I'm not so happy with my own formulation at this point, see the history. I think of a specific example of a Vitali set. It is not really constructed, it exists under the axiom of choice. And it can not be measurable since it leads to contradictions like bounds for infinite sums of equal positive terms (I missed equal in the text); I'm not sure whether we should go into all the details. The general aim is to give some explicit motivation or explanation before the formal definition using sigma-algebras is introduced (otherwise the article would be pretty technical, wouldn't it). So feel free to rework what you like. --Alex Halicz (hello) 16:08, 8 February 2007 (CST)

Link to "measurement"?

Should there be cross-links between this and measurement? Petréa Mitchell 18:53, 22 April 2007 (CDT)

Re:Link to "measurement"?

IMO no. The measurement article is about physical measurements, and little to no relation to the set theoretical measure concept.

Ragnar Schroder 14:45, 27 June 2007 (CDT)

Error

In "Basic properties", subsection "Arbitrary sequence of measurable sets" is incorrect. A correct version could be as follows.

If E_i is any sequence in Σ then

$\mu (\liminf E_{i})\leq \liminf \mu (E_{i}),$

where $\liminf E_{i}$, i.e. the lower limit of the sequence, is defined as $\bigcup _{k=1}^{\infty }\bigcap _{j=k}^{\infty }E_{j}$.

If in addition the union of all E_i is a set of finite measure then

$\mu (\liminf E_{i})\leq \liminf \mu (E_{i})\leq \limsup \mu (E_{i})\leq \mu (\limsup E_{i}),$

where $\limsup E_{i}$ (the upper limit) is equal to $\bigcap _{k=1}^{\infty }\bigcup _{j=k}^{\infty }E_{j}$.

However, for $E_{n}=[n,\infty )\subseteq \mathbb {R}$ (and Lebesgue measure μ) the right inequality fails: all $\mu (E_{i})$ are infinite but $\limsup E_{i}$ is empty.

Remarks

Introduction:

"To do this, measures may assign lengths or areas to sets that do not have a well-defined area in the traditional sense. On the other hand, it turned out that in most cases not all subsets can be assigned an area in a way which preserves the properties one expects of a measuring process.

This does for instance happen in the real line case"

— dimensions (2 and 1) are intermixed non-clearly. Maybe this is better:

"To do this, measures may assign areas to planar sets that do not have a well-defined area in the traditional sense. On the other hand, it turned out that in most cases not all subsets can be assigned an area in a way which preserves the properties one expects of a measuring process. The same holds in other dimensions (1, 3 and higher).

This does for instance happen in the real line case"

Basic properties: Increasing sequence of measurable sets:

The first property would be better in a separate subsection "Subadditivity".

"then the union of the sets En is measurable, and"

— but in other cases measurability of unions and intersections (and lower and upper limits) is not noted. I think it should not be noted here, too. Instead it could be noted, once and for all, somewhere earlier.

Basic properties: Decreasing sequence of measurable sets:

"which all have infinite measure"

— I'd say, "infinite Lebesgue measure".

Construction:

"if for all test sets"

— either remove "test" or explain it.

Sigma-finite measures:

"σ-finiteness can be compared in this respect to separability of topological spaces"

— I'd say, "of metric spaces"; for topological spaces separability is not nice (since it fails to imply second countability).

See also:

move it to a subpage.

References:

move it to a subpage, too.

Boris Tsirelson 15:47, 20 May 2010 (UTC)