Prove that : $\operatorname{inf}{\alpha >0 , \mu {f>\alpha }=0}= \operatorname{sup}{\alpha >0 , \mu {f>\alpha }>0}$

Question

Consider $(X,\Sigma, \mu)$ an arbitrary measure space

$f$ positive and measurable function

Prove that :

$$\operatorname{inf} (\{\alpha >0 , \mu (\{f>\alpha \})=0 \})=\operatorname{sup} (\{\alpha >0 , \mu (\{f>\alpha \})>0 \})$$

I don't know how I started in the prof its difficult to me Bcz I don't when and how I use the identity ?