Im looking for a distribution $P$ on $\mathbb{R}^2$ and a meassurable function $g\colon\mathbb{R}\rightarrow \mathbb{R}$ such that: $$ \int_{\mathbb{R}^2}y^2\ \mathrm dP(x,y)=\infty $$ and $$ \int_{\mathbb{R}^2}(y-g(x))^2\ \mathrm dP(x,y)<\infty. $$
I tried the density $f_P(x,y):= \mathcal{X}_{[1,\infty)^2}(x,y)/(x^2y^2)$ to fulfill the first equality, but I haven`t found a function $g$ yet to fulfill the second inequality