Lebesgue measure on borel set

In summary: This is correct. The axiom of choice is needed to make an A which is the smallest sigma-algebra containing all the open sets.
  • #1
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Can someone show me an example to clarify this statement from Royden's Real Analysis:

The Lebesgue measure restricted to the sigma-algebra of Borel sets is not complete.

Now, from the definition of a complete measure space, if B is an element of space M, and measure(B) = 0, and A subset of B, then A is an element of M.

But my understanding of the Borel sets is that it is the smallest algebra containing all the open and closed sets. Hence A would be in the Borel set, hence A would be in M.

So I'm obviously missing something.

thanks
 
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  • #2
Why is A in the set? There is nothing in what you wrote that compels a subset of a set of measure 0 to be in the Borel sigma algebra. You say it is the *smallest* sigma algebra, but behave as if it is the *largest*. A would be in the set if it could be obtained from the open (or closed) sets by operations of intersection, union, and complement.
 
  • #3
That makes sense. I guess I misinterpreted the definition.

"The collection B of Borel sets is the smallest sigma-algebra which contains all the open sets."

I read "contains" to mean: every open set A is an element of collection B.

But apparently, what "contains" means here is that every open set A is a subset of some element of B.

thanks for the help.
 
  • #4
A is *not* an open set. That's the point.
 
  • #5
Thanks again.

I think I've got it now.
 
  • #6
The collection B of Borel sets is the smallest sigma-algebra containing every open set. In particular, this means that if A is an open set, then A is an element of B. The Lebesgue measure m restricted to B is incomplete. This means that there is some A in B such that m(A) = 0 which has some set C as a subset of A such that C is not in B. The only open set with measure 0 is the empty set, and every subset of the empty set is contained in B. But there is some other element of B which isn't an open set, it has measure 0, and it also has a subset which isn't an element of B. This set A is probably some whacky set. A sigma-algebra which contains all the open sets doesn't only contain open sets, it also contains all countable intersections of open sets (which aren't necessarily open), and all countable unions of countable intersections of open sets, and all countable intersections of countable unions of countable intersections of open sets, etc.
 
  • #7
I seem to remember reading somewhere that to make such an A requires the axiom of choice, but I don't know the proof.
 

1. What is Lebesgue measure on Borel sets?

Lebesgue measure is a mathematical concept used to assign a numerical value to a set in order to measure its size or extent. It is defined on Borel sets, which are subsets of a given space that are created by taking the union or intersection of open or closed sets.

2. How is Lebesgue measure different from other measures?

Lebesgue measure differs from other measures in that it is able to assign a measure to a wider range of sets, including sets that are not necessarily simple geometric shapes. It also takes into account the concept of measure density, which allows for more precise measurements.

3. What is the importance of Lebesgue measure in mathematics?

Lebesgue measure is a fundamental concept in modern mathematics, particularly in the field of measure theory. It provides a rigorous and comprehensive way to measure the size of sets, and is used in various mathematical branches such as analysis, probability, and geometry.

4. How is Lebesgue measure calculated?

Lebesgue measure is calculated by first defining a measure on a specific set, such as the real numbers, and then extending it to a larger class of sets using a process called the Carathéodory construction. This involves defining a measure on a smaller class of sets, known as the pre-measurable sets, and then extending it to the Borel sets using certain properties.

5. What are some real-life applications of Lebesgue measure?

Lebesgue measure has various real-life applications, such as in physics, where it is used to measure the size and shape of objects in space. It is also applied in engineering, economics, and finance, where it helps in analyzing and solving complex problems involving sets and measures.

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