Let $\mu $ be a positive Borel measure on $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}$ such that $\mu \left( B\left( a,r\right) \right) \leq Cr^{n}$ for some $n\in (0,d]$ and for any ball $B\left( a,r\right) $ in $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}$. Could you help me to prove that $\int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}}\frac{1}{\left\vert x\right\vert ^{n}}d\mu \left( x\right) =\infty $?
My effort: $\int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}}\frac{1}{\left\vert x\right\vert ^{n}}d\mu \left( x\right) \geq \int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{d}\backslash B\left( 0,1\right) }\frac{1}{\left\vert x\right\vert ^{n}}% d\mu \left( x\right) =\sum_{k=0}^{\infty }\int_{B\left( 0,2^{k+1}\right) \backslash B\left( 0,2^{k}\right) }\frac{1}{\left\vert x\right\vert ^{n}}% d\mu \left( x\right) \geq \sum_{k=0}^{\infty }\frac{1}{\left( 2^{k+1}\right) ^{n}}\mu \left( B\left( 0,2^{k+1}\right) \backslash B\left( 0,2^{k}\right) \right) $.