Calculate the Integral of $\int \frac{1}{x+y}d\lambda_2(x,y)$ on $[0,1]^2$

Question

This looks like a typical exercise about Fubini´s theorem. However, f needs to be Lebesgue integrable. My prof said one can use Tonelli´s theorem to show this. Tonelli needs a function to be Lebesgue measurable and this is where I am stuck.

Answer 1

The function $\frac1{x+y}$ is continuous in $(0,1]^2$, therefore its Lebesgue measurable there.

Finally note that the set $\{0\}\times \{0\}$ have Lebesgue measure zero so we don't care what is the function there or if it is defined there because in any case the contribution of the function in this set to the value of the integral will be zero.