Simple question about the Lebesgue sigma-algebra on R^2

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In summary, the Lebesgue sigma-algebra on R², denoted by \mathfrak{L_{\mathbb{R}^2}}, is the completion of the product Borel sigma-algebra with respect to the product Lebesgue measure on R². This can also be seen as the completion of the product Lebesgue sigma-algebra on R with respect to the product Lebesgue measure on R². The conclusion of \mathfrak{B_{\mathbb{R}}}\times \mathfrak{B_{\mathbb{R}}}\subset \mathfrak{L_{\mathbb{R}}}\times \mathfrak{L_{\mathbb{
  • #1
quasar987
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Denote the Lebesgue sigma-algebra on R by [tex]\mathfrak{L_{\mathbb{R}}}[/tex], the Borel sigma-algebra on R by [tex]\mathfrak{B_{\mathbb{R}}}[/tex], the Lebesgue measure on R by [tex]\lambda[/tex].

Define the Lesbesgue sigma-algebra on R² [tex]\mathfrak{L_{\mathbb{R}^2}}[/tex] as the completion of the product Borel sigma-algebra with respect to the product Lebesgue measure on R². That is to say, [tex]\mathfrak{L_{\mathbb{R}^2}}=(\mathfrak{B_{\mathbb{R}}}\times \mathfrak{B_{\mathbb{R}}})_{\lambda \times \lambda}[/tex].

Now according to my text, the Lebesgue sigma-algebra on R² is also the completion of the product Lebesgue sigma-algebra on R with respect to the product Lebesgue measure on R². That is to say, we also have [tex]\mathfrak{L_{\mathbb{R}^2}}=(\mathfrak{L_{\mathbb{R}}}\times \mathfrak{L_{\mathbb{R}}})_{\lambda \times \lambda}[/tex].

To justify this assertion, my text proceeds to show that

[tex]\mathfrak{B_{\mathbb{R}}}\times \mathfrak{B_{\mathbb{R}}}\subset \mathfrak{L_{\mathbb{R}}}\times \mathfrak{L_{\mathbb{R}}} \subset \mathfrak{L_{\mathbb{R}^2}}[/tex]

How does the conclusion follow?
 
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  • #2
quasar987 said:
[tex]\mathfrak{B_{\mathbb{R}}}\times \mathfrak{B_{\mathbb{R}}}\subset \mathfrak{L_{\mathbb{R}}}\times \mathfrak{L_{\mathbb{R}}} \subset \mathfrak{L_{\mathbb{R}^2}}[/tex]

How does the conclusion follow?
Take completions.
 
  • #3
I see. We'll have the double inclusion

[tex]\mathfrak{L_{\mathbb{R}^2}} \subset (\mathfrak{L_{\mathbb{ R}}}\times \mathfrak{L_{\mathbb{R}}})_{\lambda \times \lambda}[/tex]

and

[tex](\mathfrak{L_{\mathbb{ R}}}\times \mathfrak{L_{\mathbb{R}}})_{\lambda \times \lambda}\subset \mathfrak{L_{\mathbb{R}^2}}[/tex]
 

1. What is the Lebesgue sigma-algebra on R^2?

The Lebesgue sigma-algebra on R^2 is a mathematical concept that helps us understand and measure the properties of sets of real numbers in two-dimensional space. It is a special type of sigma-algebra that is defined in terms of the Lebesgue measure, which is a way of assigning a numerical value to sets in R^2.

2. What are the properties of the Lebesgue sigma-algebra on R^2?

The Lebesgue sigma-algebra on R^2 has several important properties. It is closed under countable unions and intersections, meaning that if we take the union or intersection of any number of sets in the sigma-algebra, the result will still be in the sigma-algebra. It is also closed under complements, meaning that if a set is in the sigma-algebra, its complement will also be in the sigma-algebra.

3. How is the Lebesgue sigma-algebra on R^2 different from other sigma-algebras?

The Lebesgue sigma-algebra on R^2 is unique in that it is the smallest sigma-algebra that contains all the open sets in R^2. This means that it is the most natural and intuitive way to measure sets of real numbers in two-dimensional space. Other sigma-algebras may have different definitions and properties, but the Lebesgue sigma-algebra is the most commonly used in mathematical analysis and probability theory.

4. What are some applications of the Lebesgue sigma-algebra on R^2?

The Lebesgue sigma-algebra on R^2 has many practical applications in areas such as mathematical analysis, probability theory, and measure theory. It allows us to define and measure the probability of events in two-dimensional space, which is useful in understanding and predicting real-world phenomena. It also helps us analyze the properties of functions and sets in R^2, which has numerous applications in fields such as physics, economics, and engineering.

5. How is the Lebesgue sigma-algebra on R^2 related to the Lebesgue integral?

The Lebesgue sigma-algebra on R^2 is intimately connected to the Lebesgue integral, which is a way of calculating the area under a curve in two-dimensional space. The Lebesgue integral is defined in terms of the Lebesgue measure, which is based on the Lebesgue sigma-algebra. In fact, the Lebesgue integral is only defined for functions that are measurable with respect to the Lebesgue sigma-algebra. This shows the importance and relevance of the Lebesgue sigma-algebra in mathematical analysis and integration theory.

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