Is Tonelli's Theorem a Useful Tool for Determining the Existence of Integrals?

In summary, Measure theory analysis is a branch of mathematics that deals with the study of measures, which are used to quantify the size or extent of a set. It is important as it provides a foundation for other fields and helps establish the validity of mathematical theories. Key concepts include measures, measurable sets, and measurable functions. It is closely related to probability theory and has numerous applications in areas such as physics, economics, statistics, and engineering.
  • #1
HeinzBor
39
1
Homework Statement
Seeing if a funkcion is integrable
Relevant Equations
Fubinis theorem and Tonelli's theorem
Hi I am sitting with a homework problem which is to show if I can actually integrate a function. with 2D measure of lebesgue. the function is given by ##\frac{x-y}{(x+y)^2} d \lambda^2 (x,y)##.

I know that a function ##f## is integrable if ##f \in L^{1}(\mu) \iff \int |f|^{1} d \mu < \infty##.

Since ##(f \geq 0)## I can apply Tonelli's Theorem, which states that

##\int_{X \times E} f d_{\mu \times v} = \int_{X}(\int_{Y}fdv)d \mu = \int_{Y}(\int_{X}fd \mu)d v##

So my first idea was to compute both RHS and LHS and show that they do not equal if they are not measurable. But I saw that this was a complicated integral, so I was wondering if there is some other way to do it?
 
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  • #2
It is not so complicated if you split it: ##\dfrac{x-y}{(x+y)^2} = \dfrac{x}{(x+y)^2} - \dfrac{y}{(x+y)^2}##.
 
  • #3
fresh_42 said:
It is not so complicated if you split it: ##\dfrac{x-y}{(x+y)^2} = \dfrac{x}{(x+y)^2} - \dfrac{y}{(x+y)^2}##.

Thanks and then integrate both sides of tonelli's theorem ? and if they are not equal then it is not integrable because of fubini?
 
  • #4
From your deleted post:
HeinzBor said:
$\int_{[0,1] Y} \frac{x}{(x+y)^2}dv(y) - \int_{[0,1] X} \frac{y}{(x+y)^2}d \mu (x)$
For MathJax on this site, the delimiters are a pair of dollar signs ($) for standalone formulas) or a pair of hash signs (#) for inline formulas).
Using the latter, the integral above is ##\int_{[0,1] Y} \frac{x}{(x+y)^2}dv(y) - \int_{[0,1] X} \frac{y}{(x+y)^2}d \mu (x)## - that's with two # characters at the start and two more at the end.
 
  • #5
fresh_42 said:
It is not so complicated if you split it: ##\dfrac{x-y}{(x+y)^2} = \dfrac{x}{(x+y)^2} - \dfrac{y}{(x+y)^2}##.
Another possibility is ##\dfrac{x-y}{(x+y)^2} = \dfrac{x + y}{(x+y)^2} - \dfrac{2y}{(x+y)^2}##.
Also, from post #1, a minor point:
HeinzBor said:
the function is given by ##\frac{x-y}{(x+y)^2} d \lambda^2 (x,y)##.
The integrand is just this part: ##\frac{x - y}{(x + y)^2}##.
 
  • #6
Okay thank you

Okay so I gave it a try\begin{align*}

\int_{[0,1] \times [0,1]} \frac{x-y}{(x+y)^2} d \lambda^2 (x,y)

\end{align*}

Since $$f \geq 0$$ from Tonelli we have that

\begin{align*}

\int_{[0,1]} (\int_{[0,1]} \frac{x-y}{(x+y)^2} d \lambda(x)) \ d \lambda (y)\\

= \int_{[0,1]} (\int_{[0,1]} \frac{x+y}{(x+y)^2} - \frac{2y}{(x+y)^2} d \lambda(x)) \ d \lambda (y)

\end{align*}

and since $$\frac{2y}{(x+y)^2}$$ is maximum 2 we can just see it as a finite constant and then consider

\begin{align*}

&\leq \int_{[0,1]} (\int_{[0,1]} \frac{x-y}{(x+y)^2} d \lambda(x)) d \lambda (y)\\

&= \int_{[0,1]} \frac{1}{2}ln((y+1)^2) - \frac{1}{2}ln(y^2) d \lambda (y)\\

&= 2ln(2) < + \infty

\end{align*}

so f is integrable. is this correct? I hope you can help me if there's some mistakes I am still trying to learn more about the integral and measure
 
Last edited:
  • #7
HeinzBor said:
Since $f \geq 0$ from Tonelli we have that
HeinzBor said:
and since $\frac{2y}{(x+y)^2}$ is maximum 2
Again, single $ delimiters don't do anything at this site.
Please use either ## or $$ delimiters on your LaTeX stuff.
 
  • #8
Mark44 said:
Again, single $ delimiters don't do anything at this site.
Please use either ## or $$ delimiters on your LaTeX stuff.
Sorry it should be fixed now
 
  • #9
Your result looks fine to me. The integrand function is defined everywhere except at (0, 0) on the square ##[0, 1] \times [0, 1]##, so the result via Lebesque integration should be the same as with ordinary (Riemann) integration. By wolframalpha, I get a value of 0 for the integral for either order of integration.
 
  • #10
Mark44 said:
Your result looks fine to me. The integrand function is defined everywhere except at (0, 0) on the square ##[0, 1] \times [0, 1]##, so the result via Lebesque integration should be the same as with ordinary (Riemann) integration. By wolframalpha, I get a value of 0 for the integral for either order of integration.

okay thanks, so I can conclude that the integral exists?
 
  • #11
HeinzBor said:
okay thanks, so I can conclude that the integral exists?
Yes
 

Related to Is Tonelli's Theorem a Useful Tool for Determining the Existence of Integrals?

1. What is measure theory analysis?

Measure theory analysis is a branch of mathematics that deals with the study of measures, which are mathematical tools used to assign sizes or volumes to sets of objects. It provides a rigorous and systematic approach to understanding the properties of measures and their applications in various fields, such as probability theory, functional analysis, and differential equations.

2. Why is measure theory analysis important?

Measure theory analysis is important because it provides a solid foundation for many areas of mathematics and other scientific disciplines. It allows for a precise and consistent way of defining and manipulating measures, which are fundamental concepts in many fields. It also provides powerful tools for solving problems and proving theorems related to measures and their properties.

3. What are some key concepts in measure theory analysis?

Some key concepts in measure theory analysis include measure spaces, measurable sets, and measurable functions. A measure space is a set equipped with a measure, which is a function that assigns a non-negative real number to each subset of the set. Measurable sets are subsets of a measure space that can be assigned a measure, while measurable functions are functions that preserve the measure of sets in their domain.

4. How is measure theory analysis used in probability theory?

Measure theory analysis is essential in probability theory because it provides a rigorous framework for defining and manipulating probability measures. Probability measures are used to assign probabilities to events in a sample space, and they must satisfy certain properties, which can be studied using measure theory. This allows for a more precise and systematic approach to understanding and analyzing random phenomena.

5. What are some real-world applications of measure theory analysis?

Measure theory analysis has many real-world applications, including in physics, economics, and engineering. For example, in physics, it is used to study the properties of physical quantities, such as mass and energy, which can be thought of as measures. In economics, it can be used to model and analyze complex systems, such as financial markets. In engineering, it is used to design and optimize systems, such as communication networks, by considering measures of performance and efficiency.

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