Integrate of the product measure on $R^2$

Question

I have to prove this property that seems very intuitive:

Let $µ$ and $ν$ be finite Borel measures on $\mathbb{R}$ and $µ\timesν$ their product measure on $\mathbb{R^2}$. Prove:

$\int_\mathbb{R^2} (x^2+y^2)dµ \timesν(x,y) <∞$ if and only if $\int_\mathbb{R} x^2dµ(x) <∞$ and $\int_\mathbb{R} y^2dν(y) <∞$.

Sorry for the bad writting, x means the product measure.

Answer 1

This follows from the following equlities: $$\int_\mathbb{R^2} (x^2+y^2)dµ(x) \times d ν(y)= \int_\mathbb{R^2} (x^2 \times 1)dµ(x) \times d ν(y)+\int_\mathbb{R^2}(1\times y^2)dµ(x) \times d ν(y)$$ $$=\int_\mathbb{R} x^2dµ(x) \int_\mathbb{R}dv (y) +\int_\mathbb{R} d µ(x)\int_\mathbb{R} y^2dv(y)= $$ $$=\int_\mathbb{R} x^2dµ(x) \times v(\mathbb{R})+µ(\mathbb{R}) \times \int_\mathbb{R} y^2dv(y) $$