A Change of Variable/Fubini's Theorem

Question

A line in a text reads $$\int_{0}^{\infty} \mu (B(x,u^{\frac{-1}{t}}) du = t\int_{0}^{\infty} r^{-t-1} \mu (B(x,r)) dr.$$

I set $u=r^{-t}$. But then $du=-tr^{-t-1} dr$. Where is the negative?

Answer 1

When you make the change of variable, you obtain

$$\int_{\infty}^0 -tr^{-t-1} \mu(B(x,r)) dr$$

So if you want an integral between $0$ and $\infty$, the minus sign disapear