- 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?

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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