Measure Theory - The completion of R^2 under a point mass measure

In summary, the question asks for the completion of a sigma-algebra under a point mass measure concentrated at (0,0). The completion is defined as the collection of subsets for which there exist sets E and F satisfying certain conditions. The completion in this case is the original sigma-algebra itself, as any subset can be included in a larger set containing (0,0) and have a measure of zero. This process is used to add all sets of measure zero to the sigma-algebra.
  • #1
Sarcasticus
2
0
Hello;

Homework Statement


Let [tex]\mathcal{A}[/tex] be the [tex]\sigma[/tex]-algebra on [tex]\mathbb{R}^2[/tex] that consists of all unions of (possibly empty) collections of vertical lines. Find the completion of [tex]\mathcal{A}[/tex] under the point mass concentrated at (0,0).

Homework Equations



1st: Completion is defined as follows: Let [tex](X, \mathcal{A})[/tex] be a measurable space, and let [tex]\mu[/tex] be a measure on [tex]\mathcal{A}[/tex]. The completion of [tex]\mathcal{A}[/tex] under [tex]\mu[/tex] is the collection [tex]\mathcal{A}_{\mu}[/tex] of subsets A of X for which there are sets E and F in [tex]\mathcal{A}[/tex] such that
1) E [tex]\subset[/tex] A [tex]\subset[/tex] F, and
2) [tex]\mu[/tex](F-E) = 0.

2nd: A point mass measure concentrated at x is a measure [tex]\delta_x[/tex]defined on a sigma-algebra [tex]\mathcal{A}[/tex] such that, for any [tex]A \in \mathcal{A}[/tex], [tex]\delta_x(A) = 1[/tex] if [tex]x \in A[/tex] and [tex]\delta_x(A) = 0[/tex] otherwise.

The Attempt at a Solution



Here's my answer: Let [tex](\mathcal{A})_{\delta}[/tex] denote the completion of [tex]\mathcal{A}[/tex] under the pt. mass concentrated at (0,0) and let [tex]\delta[/tex] denote said measure. Then, for any set [tex]A \in \mathcal{A}[/tex], we have
[tex]A \subset A \subset A[/tex] and [tex]\delta(A-A)=0[/tex] always, so [tex]\mathcal{A} \in (\mathcal{A})_{\delta}.[/tex]
Consider any set [tex]A \in (\mathcal{A})_{\delta}[/tex]; then there exist sets E, F belonging to [tex]\mathcal{A}[/tex] such that [tex]E \subset A \subset F[/tex] and [tex]\delta(F - E) = 0[/tex]. Which means that either both E and F contain a line intersecting the origin, or neither does. This mean A will follow suit and, further, [tex]A \subset F[/tex] means that [tex]A \in \mathcal{A}[/tex] and hence [tex](\mathcal{A})_{\delta} \subset \mathcal{A}[/tex] and thus [tex]\mathcal{A} = (\mathcal{A})_{\delta}[/tex]

Except, this means the completion of any sigma algebra under a point mass measure will again be the sigma algebra. And, if this were the case, why wouldn't they just give us the general question in the first place, instead of a bunch of questions about it? (Only one displayed here.)
Hence, I think my answer mucks up somewhere.

Thanks in advance!
 
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  • #2
Think about a subset of R^2 that isn't a union of vertical lines but doesn't contain (0,0). Isn't that in the completion? I think the point is is to complete a sigma algebra of measureable sets by adding all sets of 'measure zero'.
 

1. What is "Measure Theory"?

Measure Theory is a branch of mathematics that deals with the concept of measuring sets and their properties. It provides a rigorous framework for understanding the properties of measurable sets and functions, and is widely used in various fields such as probability theory, analysis, and statistics.

2. What does it mean for R^2 to be "completed" under a point mass measure?

In Measure Theory, completion refers to the process of adding new elements to a space in order to make it more complete. In the case of R^2 being completed under a point mass measure, it means that we are adding new elements to the set of all points in the plane in order to capture the behavior of a point mass (a point with a finite mass) on that set.

3. How is the completion of R^2 under a point mass measure relevant in real-world applications?

The completion of R^2 under a point mass measure has many applications in statistics and probability theory, particularly in the study of random variables and their distributions. It allows us to model and analyze the behavior of point masses in a rigorous and consistent manner, which has numerous practical implications in fields such as physics, finance, and engineering.

4. Can you explain the concept of "measurable sets" in relation to this topic?

In Measure Theory, a measurable set is a set for which we can assign a measure (a numerical value) that captures its size and properties. In the case of the completion of R^2 under a point mass measure, measurable sets would refer to subsets of R^2 that can be assigned a measure based on the behavior of the point mass on that set.

5. Are there any limitations or challenges associated with the completion of R^2 under a point mass measure?

One limitation of this concept is that it only applies to point masses, which may not always accurately represent the behavior of real-world objects or phenomena. Additionally, the completion of R^2 under a point mass measure can be a complex mathematical process, requiring a strong understanding of Measure Theory and advanced techniques in analysis.

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