Measure theory analysis

HeinzBor

Problem 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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fresh_42

Mentor
2018 Award
It is not so complicated if you split it: $\dfrac{x-y}{(x+y)^2} = \dfrac{x}{(x+y)^2} - \dfrac{y}{(x+y)^2}$.

HeinzBor

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?

Mark44

Mentor
$\int_{[0,1] Y} \frac{x}{(x+y)^2}dv(y) - \int_{[0,1] X} \frac{y}{(x+y)^2}d \mu (x)$
Please use either $or  delimiters on your LaTeX stuff. HeinzBor Again, single  delimiters don't do anything at this site. Please use either$ or  delimiters on your LaTeX stuff.
Sorry it should be fixed now

Mark44

Mentor
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.

HeinzBor

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?

Mark44

Mentor

"Measure theory analysis"

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