This is a question from past comp exam,
Let ${(g_n)_n }$ be a real continuous funtion with support in $(\frac{1}{n+1}, \frac{1}{n})$, such that $\int_0^1 g_n(t)dt =1$, for $n=1,2,3....$ Now define $ f(x,y)= \sum _n^\infty [g_n(x)-g_{n+1}(x)] g_n(y)$ on $[0,1]*[0,1]$.
I am trying to prove this function is not integrable on $[0,1]*[0,1]$. What I know is I need to prove that the absolute value of the function $f(x,y)$ is not integrable. on $[0,1]*[0,1]$. showing that the sum actually goes to infinity. I would love to see if someone there who like to prove this fact rigorously. In addition why this fact does not contradict the hypothesis of the Tonelli's and Fubini's Theorem. Thank you in advance.