Talk:Outer measure

Comments

It turns out the second and third requirements together for all sets are incompatible conditions

I'm not sure this is true. Counting measure is a counterexample. I think if you include the first requirement as well (which should imply translation invariance) then the measure cannot be defined for all subsets. Also, is outer measure something which can be defined on (the power set of) any set?, or just Euclidean spaces? Revolver 17:17, 8 Mar 2004 (UTC)

Counting measure is a counterexample + it is translation invariant, but, of course, it does not measure length, area, volume, et. c. -- Leocat 18:36, 31 October 2006 (UTC)

Measurable sets

In the section "Formal definitions" you state: "A subset E of X is φ-measurable iff for every subset A of X : $\varphi (A)=\varphi (A\cap E)+\varphi (A\setminus E).$"

What if the set A is not measurable?

-- Leocat 18:36, 31 October 2006 (UTC)

The definition (due to Caratheodory) explicitly quantifies over all subsets of X. What's the problem?--CSTAR 18:45, 31 October 2006 (UTC)

If the set A is not measurable and contained in E, then $\varphi (A\cap E)$ is not defined. -- Leocat 11:07, 1 November 2006 (UTC)

The definition says

An outer measure is a function defined on all subsets of a set X

$\varphi :2^{X}\rightarrow [0,\infty ]$

such that some additional conditions (countable subadditivity etc.) hold. Therefore $\varphi (A\cap E)$ is defined for any subset A of X. This is the standard definition; please look at any of the cited refernces. --CSTAR 15:06, 1 November 2006 (UTC)

However, it may be useful to know that if the outer measure is defined using method I, then it suffices to test only A from C. — Emil J. 16:17, 6 August 2009 (UTC)

A mistake?

Let's see the sentence "Let X be a set, C a family of subsets of X which contains the empty set and p an extended real valued function on C which vanishes on the empty set" in the first of the section "Construction of outer measures". It seems that the words "p an extended real valued function" should be read as "p an extended nonnegative real valued function". —Preceding unsigned comment added by Ptnam373 (talk • contribs) 20:51, 7 July 2009 (UTC)

Right. — Emil J. 16:17, 6 August 2009 (UTC)

translation invariance

I think that really all 4 conditions are needed in the "motiviation" section. The zero function satisfies everything except the first condition, and if there is a real-valued measurable cardinal then Lebesgue measure can be extended to a total measure. --Aleph4 (talk) 15:53, 11 August 2010 (UTC)

Definition

I noticed that there is probably some redundancy in the definition. I suspect that one can replace the three requirements by the following:

• Countable subadditivity: Whenever {U_j} is a countable collection of sets such that

$A\subset \bigcup _{j}U_{j},$

then

$\varphi (A)\leq \sum _{j}\varphi (U_{j}).$

This definition is more economical, and has the advantage that it can be sourced to an excellent reference, namely Geometric Measure Theory by Herbert Federer. Economy in definitions is probably appropriate when it comes to "lowlevel mathematics". (This definition is obtained through a slight generalization of Caratheodory's original definition; Federer omits Caratheodory's requirement of additivity for disjoint sets.) YohanN7 (talk) 15:34, 16 July 2013 (UTC)

Confusing claim about area

In the section "Measurability of sets relative to an outer measure", the article first gives the definition of measurability:

One says that a subset $E$ of $X$ is $\mu$-measurable (sometimes called Carathéodory-measurable relative to $\mu ,$ after the mathematician Carathéodory) if and only if

$$\mu (A)=\mu (A\cap E)+\mu (A\setminus E)$$

for every subset $A$ of $X.$

It then says:

One might then expect that every subset of the plane would be deemed "measurable," following the expected principle that

$$\operatorname {area} (A\cup B)=\operatorname {area} (A)+\operatorname {area} (B)$$

whenever $A$ and $B$ are disjoint subsets of the plane.

I am having trouble seeing how these two formulas relate to one another. Does the second one imply the first somehow? If so it would be great to include a more precise statement and a proof, since this seems like it would give a really nice intuitive motivation for the definition.

(Maybe something along the lines of "if $\mu (A)+\mu (B)=\mu (A\cup B)$ for all disjoint A and B, then all sets are $\mu$-measurable"? Or even "A is $\mu$-measurable if and only if for every B disjoint from A we have $\mu (A)+\mu (B)=\mu (A\cup B)$"? But I don't immediately see why either of these would be true.)

In its present form I think this claim is just confusing, as it implies some kind of relationship between the area rule and Carathéodory-measurability but doesn't really explain what it is.

Nathaniel Virgo (talk) 08:13, 24 June 2023 (UTC)