Borel measures on $\mathbb{R}$ questions

Question

I am reading a textbook and need some help. First it mentions that we can find a Borel measure such that $\int_\mathbb{R} x^2 \mu(x)<\infty$ but $\int_\mathbb{R} x \mu(x)=\infty$. This seems logical to me but I cannot find an example. Second, is there such a measure that the opposite is true ($\int_\mathbb{R} x^2 \mu(x)=\infty$ but $\int_\mathbb{R} x \mu(x)<\infty$)? Thanks for the help.

Answer 1

For the first example, take $d\mu(x)=u(x)\, dx$, where $u(x)=0$ if $x\not\in (0,1)$ and $u(x)=\frac1{x^2}$ if $x\in (0,1)$. Then $\int_{\mathbb R} x^2 d\mu(x)=\int_0^1 dx=1$ and $\int_{\mathbb R} x\, d\mu(x)=\int_0^1\frac{dx}x=\infty$.

For the second example, take $d\mu(x)= v(x)\, dx$, where $v(x)=0$ if $x<1$ and $v(x)=\frac1{x^3}$ if $x\geq 1$.